A Quantitative Stability Theorem for Convolution on the Heisenberg Group

TL;DR

This paper establishes a quantitative stability result for convolution on the Heisenberg group, showing near-maximizers are close to Gaussian triples, extending understanding of convolution inequalities in non-commutative settings.

Contribution

It provides a quantitative stability theorem for convolution on the Heisenberg group, characterizing near-maximizers as close to Gaussian triples, which was previously only qualitatively understood.

Findings

Near-maximizers are close to Gaussian triples after symmetry adjustments.

The expansion method effectively quantifies the stability of convolution inequalities.

The result parallels known Euclidean space results in a non-commutative setting.

Abstract

Although convolution on Euclidean space and the Heisenberg group satisfy the same $L^p$ bounds with the same optimal constants, the former has maximizers while the latter does not. However, as work of Christ has shown, it is still possible to characterize near-maximizers. Specifically, any near-maximizing triple of the trilinear form for convolution on the Heisenberg group must be close to a particular type of triple of ordered Gaussians after adjusting by symmetry. In this paper, we use the expansion method to prove a quantitative version of this characterization.

Equations

$${\mathcal{T}}_{{\mathbb{R}}^d}(\mathbf{f}):={\iint }_{{\mathbb{R}}^d\times {\mathbb{R}}^d}{f}_1(x)f_2(y)f_3(-x-y)dxdy$$

$$|{\mathcal{T}}_{{\mathbb{R}}^d}(\mathbf{f})|\le \prod _{j=1}^3\Vert f_j{\Vert }_{p_j}$$

$$|{\mathcal{T}}_{{\mathbb{R}}^d}(\mathbf{f})|\le {\mathbf{A}}_{\mathbf{p}}^d\prod _{j=1}^3\Vert f_j{\Vert }_{p_j},$$

$${\mathbf{A}}_{\mathbf{p}}=\prod _{j=1}^3{p}_j^{1/(2p_j)}/(p_j^{\prime }{)}^{1/(2p_j^{\prime })}$$

$${\text{dist}}_{\mathbf{p}}(\mathcal{O}(\mathbf{f}),\mathbf{g}):=\underset{\mathbf{h}\in {\mathcal{O}}_C(\mathbf{f})}{\inf }\underset{j}{\max }\Vert h_j-g_j{\Vert }_{p_j}.$$

$$|{\mathcal{T}}_{{\mathbb{R}}^d}(\mathbf{f})|\le \left({\mathbf{A}}_{\mathbf{p}}^d-c{\text{dist}}_{\mathbf{p}}(\mathcal{O}(\mathbf{f}),\mathbf{g}{)}^2\right)\prod _j\Vert f_j{\Vert }_{p_j}.$$

$$z\cdot z^{\prime }=(x,t)\cdot (x^{\prime },t^{\prime }):=(x+x^{\prime },t+t^{\prime }+\sigma (x,x^{\prime })),$$

$$\sigma (x,x^{\prime }):=\sum _{j=1}^d{x}_j{x}_{j+d}^{\prime }-x_{j+d}{x}_j^{\prime }.$$

$${\mathcal{T}}_{{\mathbb{H}}^d}(\mathbf{f})={\iint }_{{\mathbb{H}}^d\times {\mathbb{H}}^d}{f}_1(z_1)f_2(z_2)f_3(z_2^{-1}{z}_1^{-1})d{z}_{1}d{z}_2.$$

$$|{\mathcal{T}}_{{\mathbb{H}}^d}(\mathbf{f})|\le {\mathbf{A}}_{\mathbf{p}}^{2d+1}\prod _{j=1}^3\Vert f_j{\Vert }_{p_j},$$

$$\max \{a,a^{1/2},b\}\Vert L^{-1}\Vert <ϵ.$$

$$\Vert f_j-G_j{\Vert }_{p_j}<C\sqrt{\delta }\Vert f_j{\Vert }_{p_j}\text{ for }j\in \{1,2,3\}.$$

$$\mathcal{T}(\mathbf{f},A,b):=\iint f_1(z_1)f_2(z_2)f_3(-z_1-z_2-e_{2d+1}\sigma (A{x}_1,A{x}_2))e^{ib\sigma (A{x}_1,A{x}_2)}d{z}_{1}d{z}_2,$$

$$\begin{gathered} \Phi (M_{-b{A}^{T}JA(U_2-U_3)}{\tilde{\tau }}_{w_1}^A{\tau }_{u_1}^A{f}_1,M_{-b(A(U_1-U_2){)}^{T}JA}{\tilde{\tau }}_{w_2}^A{\tau }_{u_2}^A{f}_2,{\tilde{\tau }}_{w_3}^A{\tau }_{u_3}^A{f}_3,A,b) \\ =\Phi (f_1,f_2,f_3,A,b). \end{gathered}$$

$$\Phi ({\tilde{\tau }}_{w_1}^{Id}{\tau }_{u_1}^{Id}{f}_1,{\tilde{\tau }}_{w_2}^{Id}{\tau }_{u_2}^{Id}{f}_2,{\tilde{\tau }}_{w_3}^{Id}{\tau }_{u_3}^{Id}{f}_3,A,b)=\Phi (f_1,f_2,f_3,A,b),$$

$$\begin{gathered} {\text{dist}}_{\mathbf{p}}(\tilde{\mathcal{O}}(\mathbf{f},A,b),(\mathbf{g},0,0){)}^2:= \\ \underset{(\mathbf{h},M,r)\in \tilde{\mathcal{O}}(\mathbf{f},A,b)}{\inf }\underset{j}{\max }\Vert h_j-g_j{\Vert }_{p_j}^2+\Vert M^{T}JM{\Vert }^2+r^2\Vert M^{T}JM{\Vert }^2. \end{gathered}$$

$$|\mathcal{T}(\mathbf{f},A,b)|\le ({\mathbf{A}}_{\mathbf{p}}^{2d+1}-c{\text{dist}}_{\mathbf{p}}(\tilde{\mathcal{O}}(\mathbf{f},A,b),(\mathbf{g},0,0){)}^2)\prod _j\Vert f_j{\Vert }_{p_j}.$$

$${\text{dist}}_{\mathbf{p}}(\tilde{\mathcal{O}}(\mathbf{f},Id,0),(\mathbf{g},0,0))<C{\delta }^{1/2}$$

$$\Vert e^{ibt}(\Psi f_j)\circ A-g_j\Vert =O({\delta }^{1/2})\Vert g_j{\Vert }_{p_j},$$

$$\Vert A^{T}JA\Vert =\underset{S\in Sp(2d)}{\inf }\Vert S^{-1}\circ A{\Vert }^2,$$

$$\Vert f_j-{\Psi }^{-1}(e^{-ibt}{g}_j\circ L^{-1})\circ S^{-1}{\Vert }_{p_j}=O({\delta }^{1/2})\Vert {\Psi }^{-1}(e^{-ibt}{g}_j\circ L^{-1})\circ S^{-1}{\Vert }_{p_j}=O({\delta }^{1/2})\Vert f_j{\Vert }_{p_j}.$$

$${\text{dist}}_{\mathbf{p}}(\tilde{\mathcal{O}}(\mathbf{f},A,b),(\mathbf{g},0,0))<ϵ(\delta ).$$

$$\Vert f_j-G_j{\Vert }_{p_j}<ϵ(\delta )\Vert f_j{\Vert }_{p_j}\text{ for }j\in \{1,2,3\}.$$

$$\frac{\Vert F_j-\Psi (e^{irt}{g}_j\circ L){\Vert }_{p_j}}{\Vert F_j{\Vert }_{p_j}}<ϵ(\delta )$$

$$\frac{\Vert e^{-irt}{\Psi }^{-1}{F}_j\circ L^{-1}-g_j{\Vert }_{p_j}}{\Vert e^{-irt}{\Psi }^{-1}{F}_j\circ L^{-1}{\Vert }_{p_j}}<ϵ(\delta ).$$

$${\text{dist}}_{\mathbf{p}}(\tilde{\mathcal{O}}(\mathbf{f},A,b),(\mathbf{g},0,0){)}^2\le$$

$$+(1+r^2)\underset{S\in Sp(2d)}{\inf }\Vert S\circ S_0^{-1}\circ L^{-1}{\Vert }^4$$

$$\le$$

$$\begin{gathered} \frac{\mathcal{T}(\mathbf{g}+\mathbf{f},A,b)}{{\prod }_j\Vert g_j+f_j{\Vert }_{p_j}}=\frac{\mathcal{T}(\mathbf{g}+\mathbf{f},A,b)-\mathcal{T}(\mathbf{g}+\mathbf{f},A,0)}{{\prod }_j\Vert g_j+f_j{\Vert }_{p_j}} \\ +\frac{\mathcal{T}(\mathbf{g}+\mathbf{f},A,0)-\mathcal{T}(\mathbf{g}+\mathbf{f},0,0)}{{\prod }_j\Vert g_j+f_j{\Vert }_{p_j}}+\frac{\mathcal{T}(\mathbf{g}+\mathbf{f},0,0)}{{\prod }_j\Vert g_j+f_j{\Vert }_{p_j}} \end{gathered}$$

$$f_{j,\sharp}=\left\{\begin{array}[]{lr}f_{j}(x)\text{ if }|f_{j}(x)|\leq\eta g_{j}(x)\\ 0\text{ otherwise, }\end{array}\right.$$

$$\begin{gathered} \frac{\mathcal{T}(\mathbf{g}+\mathbf{f},A,0)-\mathcal{T}(\mathbf{g}+\mathbf{f},0,0)}{{\prod }_j\Vert g_j+f_j{\Vert }_{p_j}}\le -C\Vert A^{T}JA{\Vert }^2 \\ +o((\Vert f{\Vert }_{\mathbf{p}}+\Vert A^{T}JA\Vert {)}^2)+O(\Vert {\mathbf{f}}_{♯}{\Vert }_{\mathbf{p}}\Vert {\mathbf{f}}_{♭}{\Vert }_{\mathbf{p}}+\Vert {\mathbf{f}}_{♭}{\Vert }_{\mathbf{p}}^2). \end{gathered}$$

Full text

Keywords: Heisenberg group, quantitative stability, sharp constants

1. Introduction

For triples of functions $\mathbf{f}=(f_{1},f_{2},f_{3})$ with $f_{j}:\mathbb{R}^{d}\to\mathbb{C}$, let

[TABLE]

denote the trilinear form of convolution on $\mathbb{R}^{d}$.

In dual form, Young’s convolution inequality states that for any triple of exponents $\mathbf{p}=(p_{1},p_{2},p_{3})\in[1,\infty]^{3}$ with $\sum_{j=1}^{3}p_{j}^{-1}=2$,

[TABLE]

for all $f_{j}\in L^{p_{j}}$ ($1\leq j\leq 3$). (In this scenario, we will write $\mathbf{f}\in L^{\mathbf{p}}$.) Such $\mathbf{p}$ will be deemed admissible.

Beckner [1] and Brascamp and Lieb [4] established the stronger statement that

[TABLE]

where

[TABLE]

and $p^{\prime}$ is the conjugate exponent to $p$. Moreover, $\mathbf{A}_{\mathbf{p}}^{d}$ is the optimal constant in (1.1).

Brascamp and Lieb [4] showed that equality is attained in (1.1) precisely when $\mathbf{f}$ is the particular ordered triple of Gaussians $\mathbf{g}=(g_{1},g_{2},g_{3})$ with $g_{j}(z)=e^{-\pi p_{j}^{\prime}|z|^{2}}:=e^{-\gamma_{j}|z|^{2}}$, or the orbit of $\mathbf{g}$ under the symmetries of the operator: scaling, translation, modulation, and diagonal action of the general linear group $Gl(d)$. The non-trivial part of this result is the uniqueness of maximizers up to symmetry; the set of maximizers must be invariant under symmetries of the operator since they do not change the ratio of the left hand side of (1.1) to the right hand side.

If equality is nearly attained in (1.1) for a particular triple of functions $\mathbf{f}$, then one would like to say that $\mathbf{f}$ is close to $\mathbf{g}$. As stated, this is false, as the symmetries of convolution may be used to send an $\mathbf{f}$ which is close to $\mathbf{g}$ to another near-maximizing triple, far from $\mathbf{g}$. For example, if $f_{j}=g_{j}+\delta\phi$ for $\phi\in C_{c}^{\infty}(\mathbb{R}^{d})$ and small $\delta>0$, then $\mathbf{f}$ is close to $\mathbf{g}$; yet, if $f_{j}^{\prime}(x)=f_{j}(10^{5}x)$, $\mathcal{T}_{\mathbb{R}^{d}}(\mathbf{f}^{\prime})/\prod_{j}\|f_{j}^{\prime}\|_{p_{j}}=\mathcal{T}_{\mathbb{R}^{d}}(\mathbf{f})/\prod_{j}\|f_{j}\|_{p_{j}}$ while $\mathbf{f}^{\prime}$ is far from $\mathbf{g}$. (A formal definition of closeness will be given shortly.) However, one may obtain a positive result of this type, provided one merely conclude some element in the orbit of $\mathbf{f}$ is close to $\mathbf{g}$.

Let $\mathcal{O}(\mathbf{f})$ denote the orbit of a triple of functions $\mathbf{f}=(f_{1},f_{2},f_{3})$ under the aforementioned symmetries. Define the distance function

[TABLE]

A recent result of Christ [6] states:

Theorem 1.1**.**

Let $K$ be a compact subset of $(1,2)^{3}$. For each $d\geq 1$, there exists $c>0$ such that for all admissible $\mathbf{p}\in K$ and all $\mathbf{f}\in L^{\mathbf{p}}(\mathbb{R}^{d})$,

[TABLE]

A particular rephrasing of the conclusion of Theorem 1.1 states there exists $C>0$ such that if $|\mathcal{T}_{\mathbb{R}^{d}}(\mathbf{f})|\geq(1-\delta)\mathbf{A}_{\mathbf{p}}^{d}\prod_{j}\|f_{j}\|_{p_{j}}$, then $\text{dist}_{\mathbf{p}}(\mathcal{O}_{C}(\mathbf{f}),\mathbf{g})<C\sqrt{\delta}$. As originally stated in [6], the distance is between $\mathbf{f}$ and the manifold of maximizing triples (i.e., $\mathcal{O}(\mathbf{g})$), though the two definitions are easily shown to be equivalent after proper rescaling.

The purpose of this paper is to prove an analogue of Theorem 1.1 for the Heisenberg group.

The Heisenberg group $\mathbb{H}^{d}$ is the set $\mathbb{R}^{2d+1}$ identified as $\{z=(x,t):x\in\mathbb{R}^{2d},t\in\mathbb{R}\}$ with the group operation

[TABLE]

where $\sigma$ is the symplectic form on $\mathbb{R}^{2d}$ defined by

[TABLE]

In $\mathbb{H}^{d}$, the inverse of $(x,t)$ is $(-x,-t)$ and the Haar measure is Lebesgue measure on $\mathbb{R}^{2d+1}$.

Define the trilinear form for convolution on $\mathbb{H}^{d}$ by

[TABLE]

It was shown by Klein and Russo [8] and Beckner [2] that

[TABLE]

where $\mathbf{A}_{\mathbf{p}}^{2d+1}$ is optimal. (This is the same $\mathbf{A}_{\mathbf{p}}^{2d+1}$ as is defined in (1.2)). Furthermore, Beckner observed that there are no maximizers of (1.4).

Consider the example $f_{j}(x,t)=e^{-\gamma_{j}(\lambda|x|^{2}+\lambda^{-1}t^{2}+i\lambda^{-1}t)}$, where $\lambda\to\infty$. Viewed as functions on $\mathbb{R}^{2d+1}$, $\mathbf{f}$ is a maximizing triple for convolution for all $\lambda$. However, one may check by computation that $\mathbf{f}$ is a maximizing sequence for convolution on $\mathbb{H}^{d}$ (that is, $\mathcal{T}_{\mathbb{H}^{d}}(\mathbf{f})/\prod_{j}\|f_{j}\|_{p_{j}}\to\mathbf{A}_{\mathbf{p}}^{d}$), yet equality is not attained in (1.4) for any $\lambda$. Furthermore, the limit of $\mathbf{f}$ as $\lambda\to\infty$ does not exist in $L^{\mathbf{p}}(\mathbb{H}^{d})$. While this is not a proof of Beckner’s observation, it does provide a useful heuristic.

What accounts for this difference between $\mathbb{R}^{2d+1}$ and $\mathbb{H}^{d}$? One explanation is that on $\mathbb{R}^{2d+1}$, the diagonal action of $Gl(2d+1)$ is a symmetry for convolution, allowing one to “return” $\lambda|x|^{2}+\lambda^{-1}t^{2}$ to the $|x|^{2}+t^{2}$ found in the standard maximizing triple $\mathbf{g}$. Furthermore, the modulation symmetry allows one to remove the oscillatory factor.

On $\mathbb{H}^{d}$ however, the symmetries of convolution do not include modulation in the $t$ variable nor the entirety of $Gl(2d+1)$. In some sense, these “missing symmetries” are the only obstacle to the existence of maximizers and all maximizing triples for convolution on $\mathbb{H}^{d}$ are close to a triple of similar form after adjusting by the appropriate symmetries (see the work of Christ, [7]). The goal of this paper is to provide quantitative bounds for this closeness.

To state our main result requires a little more background.

Let $Sp(2d)$ denote the symplectic group on $\mathbb{R}^{2d}$, the set of matrices $S$ such that $\sigma(Sx,Sy)=\sigma(x,y)$ for all $x,y\in\mathbb{R}^{2d}$ with the group operation of matrix multiplication.

Formally, by a symmetry of $\mathcal{T}_{\mathbb{H}^{d}}$, we mean an operation on $\mathbf{f}$ which preserves the ratio $|\mathcal{T}_{\mathbb{H}^{d}}(\mathbf{f})|/\prod_{j}\|f_{j}\|_{p_{j}}$. For $\mathbb{H}^{d}$, the symmetries of interest are:

• •

$f_{j}\mapsto a_{j}f_{j}$ for $a_{j}\in\mathbb{C}\setminus\{0\}$. (Scaling)

• •

$f_{j}(x,t)\mapsto f_{j}(rx,r^{2}t)$ for $r\in(0,\infty)$. (Dilation)

• •

$f_{j}(z)\mapsto f_{j}(u_{j}zw_{j})$ with $w_{1}=u_{2}^{-1},w_{2}=u_{3}^{-1}$, and $w_{3}=u_{1}^{-1}$. (Translation)

• •

$f_{j}(x,t)\mapsto f_{j}(Sx,t)$ for $S\in Sp(2d)$. (Diagonal Action of the Symplectic Group)

• •

$f_{j}(x,t)\mapsto f_{j}(x,t+\varphi(x))$, where $\varphi:\mathbb{R}^{2d}\to\mathbb{R}$ is linear. (Shear Transformation)

• •

$f_{j}(x,t)\mapsto e^{iu\cdot x}f_{j}(x,t)$ for $u\in\mathbb{R}^{2d}$. (Modulation in $x$)

Let $\Sigma(\mathcal{T}_{\mathbb{H}^{d}})$ denote the group generated by the above symmetries. Following [7], we define a canonical $\epsilon$-diffuse Gaussian to be a function of the form $G(x,t)=e^{-|Lx|^{2}-at^{2}+ibt}$, where $a>0,b\in\mathbb{R},L\in Gl(2d)$, and

[TABLE]

Furthermore, given admissible $\mathbf{p}$, a triple of canonical $\epsilon$-diffuse Gaussians of the form $G_{j}=e^{-|L_{j}x|^{2}-a_{j}t^{2}+ib_{j}t}$ is said to be $\mathbf{p}$-admissible if there exist $L,a$, and $b$ such that $L_{j}=\gamma_{j}^{1/2},a_{j}=\gamma_{j}a$, and $b_{j}=b$ for $1\leq j\leq 3$.

Lastly, we say an ordered triple $\mathbf{G}=(G_{1},G_{2},G_{3})$ of Gaussians is $\epsilon$-diffuse and $\mathbf{p}$-compatible if there exists $\Psi\in\Sigma(\mathcal{T}_{\mathbb{H}^{d}})$ and a $\mathbf{p}$-compatible ordered triple $\tilde{\mathbf{G}}=(\tilde{G}_{1},\tilde{G}_{2},\tilde{G}_{3})$ of canonical $\epsilon$-diffuse Gaussians such that $\mathbf{G}=\Psi\tilde{\mathbf{G}}$ for $1\leq j\leq 3$.

Our main result is the following:

Theorem 1.2**.**

Let $d\geq 1$ and $K\subset(1,2)^{3}$ be compact. Then, there exists a $C>0$ with the following property. Let $\mathbf{p}\in K$ be admissible, $\mathbf{f}\in L^{\mathbf{p}}(\mathbb{H}^{d})$, and $\|f_{j}\|_{p_{j}}\neq 0$ for all $1\leq j\leq 3$. Let $\delta\in(0,1)$ and suppose that $|\mathcal{T}_{\mathbb{H}^{d}}(\mathbf{f})|\geq(1-\delta)\mathbf{A}_{\mathbf{p}}^{2d+1}\prod_{j}\|f_{j}\|_{p_{j}}$. Then there exists a $\mathbf{p}$-compatible $C\sqrt{\delta}$-diffuse ordered triple of Gaussians $\mathbf{G}=(G_{1},G_{2},G_{3})$ such that

[TABLE]

The exponent 2 found in (1.5) is sharp.

Prior work of Christ [7] establishes a qualitative stability theorem of a similar form. This result is of the same form as Theorem 1.2, yet refers to an undetermined function $\epsilon(\delta)$ satisfying $\lim_{\delta\to 0}\epsilon(\delta)=0$ in place of $C\sqrt{\delta}$. We state this result as Theorem 2.3 and use it to reduce to small perturbations in Section 2.

Also in Section 2, we will develop a translation scheme between convolution on the Heisenberg group and a generalized operator which will allow us to prove Theorem 1.2 through the expansion method of Bianchi and Egnell [3].

In Sections 3 and 4, we compute some terms of the expansion. Section 5 determines what is needed to apply a sharpened form of Young’s inequality due to Christ to handle the remaining term of the expansion. In Section 6, we prove a balancing lemma to attain these conditions, allowing us to combine all the terms and conclude the proof of Theorem 1.2 in Section 7.

In Section 8, we establish some variants of Theorem 1.2 in cases where at least one $p_{j}$ is greater than or equal to 2, as in [6].

Acknowledgment: The author would like to thank Michael Christ for the suggestion of the problem and some helpful conversations.

2. Translation Into a Distance

To prove Theorem 1.2, we will use the expansion method. The first obstacle in performing the expansion is that, as previously discussed, there are no maximizers for convolution on the $\mathbb{H}^{d}$. Our solution is to use the maximizers for convolution on $\mathbb{R}^{2d+1}$ for comparison, continuously varying the group structure between that of the two spaces. This leads to another obstacle, in that the symmetry groups for convolution on $\mathbb{H}^{d}$ and $\mathbb{R}^{2d+1}$ differ. This is a problem because with the differing symmetry groups, a near-maximizer for convolution on $\mathbb{H}^{d}$ such as $f_{j}(x,t)=e^{-\gamma_{j}(\lambda|x|^{2}+\lambda^{-1}t^{2}+i\lambda^{-1}t)}$ cannot be sent to a small neighborhood of the fixed maximizing triple $g_{j}(z)=e^{-\gamma_{j}|z|^{2}}$ under the symmetries for $\mathbb{H}^{d}$.

To resolve this second issue, we introduce a new functional which generalizes the trilinear forms for convolution on both $\mathbb{R}^{2d+1}$ and $\mathbb{H}^{d}$. This new functional effectively allows for a more flexible group structure, so general elements of $Gl(2d+1)$ may act as symmetries by changing this group structure. Similar ideas were applied to the case of twisted convolution by the author in [9].

The generalized functional is:

[TABLE]

where $\mathbf{f}\in L^{\mathbf{p}}(\mathbb{R}^{2d+1})$, $A$ is a $(2d)\times(2d)$ matrix, $b\in\mathbb{R}$, and $e_{2d+1}=(0,...,0,1)\in\mathbb{R}^{2d+1}$. $(A,b)$ will be referred to as the attached parameters.

Since one may view $\mathcal{T}$ as a convolution-like operator with varying group structure, it will be helpful to use $\cdot_{A}$ to denote the group operation $(x,t)\cdot_{A}(x^{\prime},t^{\prime}):=(x+x^{\prime},t+t^{\prime}+\sigma(Ax,Ax^{\prime}))$.

Let $\tau^{A}_{u}f(z)=f(u\cdot_{A}z)$ and $\tilde{\tau}^{A}_{w}f(z)=f(z\cdot_{A}w)$ to represent left and right translation, respectively. Modulation will be represented by the notation $M_{\xi}f(z)=e^{iz\cdot\xi}f(z)$. We will write $D_{r}f(x,t)=f(rx,r^{2}t)$ for dilation. Through slight abuse of notation, we will often write $f\circ A(x,t)=f(Ax,t)$ for linear maps $A:\mathbb{R}^{2d}\to\mathbb{R}^{2d}$ and $\mathbf{f}\circ A$ to denote $(f_{1}\circ A,f_{2}\circ A,f_{3}\circ A)$.

By a symmetry of $\mathcal{T}$, we mean an operation on $(\mathbf{f},A,b)$ which preserves $\Phi(\mathbf{f},A,b):=|\mathcal{T}(\mathbf{f},A,b)|/\left(\prod_{j=1}^{3}\|f_{j}\|_{p_{j}}\right)$. Here, the relevant symmetries are:

• •

$\Phi((a_{1}f_{1},a_{2}f_{2},a_{3}f_{3}),A,b)=\Phi((f_{1},f_{2},f_{3}),A,b)$ for $a_{j}\in\mathbb{C}\setminus\{0\}$. (Scaling)

• •

For $u_{j}=(U_{j},U_{j}^{\prime}),w_{j}=(W_{j},W_{j}^{\prime})\in\mathbb{R}^{2d}\times\mathbb{R}$ satisfying $w_{1}=u_{2}^{-1},w_{2}=u_{3}^{-1}$, and $w_{3}=u_{1}^{-1}$,

[TABLE]

(Translation-Modulation)

• •

$\Phi(M_{\xi}f_{1},M_{\xi}f_{2},M_{\xi}f_{3},A,b+\xi_{2d+1})=\Phi(f_{1},f_{2},f_{3},A,b)$ for $\xi\in\mathbb{R}^{2d+1}$. (Modulation)

• •

$\Phi(\mathbf{f}\circ L,AL,b)=\Phi(\mathbf{f},A,b)$ for $A\in Gl(2d)$. (Diagonal Action of $Gl(2d)$)

• •

$\Phi(D_{r}f_{1},D_{r}f_{2},D_{r}f_{3},A,r^{2}b)=\Phi(f_{1},f_{2},f_{3},A,b)$ for $r\in(0,\infty)$. (Dilation)

• •

Let $\varphi:\mathbb{R}^{2d}\to\mathbb{R}$ be linear and let $g_{j}=f_{j}(x,t+\varphi(x))$. Then, $\Phi(\mathbf{g},A,b)=\Phi(\mathbf{f},A,b)$. (Sheer Transformation)

Note that $A$ changes precisely under $Gl(2d)$ symmetries, and $b$ changes precisely under modulations in the $(2d+1)$-st coordinate and dilation.

While some of the above symmetries may appear complicated, we will see shortly that for our purpose they may usually be applied in the special case $A=Id,b=0$, simplifying their expressions; here and throughout, $Id$ refers to the $(2d)\times(2d)$ identity matrix. (For instance, the translation-moldulation symmetry becomes

[TABLE]

where $\tau^{Id}$ and $\tilde{\tau}^{Id}$ represent the usual translation on $\mathbb{H}^{d}$.) This is because the definition of orbit will allow for rather restricted use of symmetries.

At this point, one may expect to prove a direct analogue of Theorem 1.1 for the operator $\mathcal{T}(\mathbf{f},A,b)$. While this is possible, we desire something a little stronger to recover Theorem 1.2. The $c\sqrt{\delta}$-diffuse Gaussians of Theorem 1.2 are not obtained through the action of any symmetry of $\mathcal{T}(\mathbf{f},A,b)$ on $\mathbf{g}$; rather, symmetries of $\mathcal{T}_{\mathbb{H}^{d}}$ are applied only after symmetries of $\mathcal{T}(\mathbf{f},A,b)$ which are not symmetries of $\mathcal{T}(\mathbb{H}^{d})$. Thus, we must define an alternative to the usual notion of orbit.

Let $\mathfrak{G}_{0}$ be the group of symmetries of $\mathcal{T}(\mathbf{f},A,b)$ generated by $Gl(2d)$ and modulation in the $t$ variable and let $\mathfrak{G}_{1}$ be the group generated by the remaining symmetries on the list, along with the diagonal action of $Sp(2d)$. Note that $\mathfrak{G}_{1}$ is in one-to-one correspondence with $\Sigma(\mathcal{T}_{\mathbb{H}^{d}})$ when $A=Id$ and $b=0$. While $\Psi\in\mathfrak{G}_{j}$ ($j=0,1$) is defined as acting on a tuple of the form $(\mathbf{f},A,b)$, we will often write $\Psi f_{j}$ to denote its action on a particular function. In such a scenario, we will attempt to be particularly clear on what $\Psi$ does to $f_{j}$, given the action may depend on the attached parameters.

Let $\tilde{\mathcal{O}}(\mathbf{f},A,b)$ denote the set of elements of the form $\Psi_{0}\Psi_{1}((e^{-ibt}\mathbf{f})\circ A^{-1},Id,0)$ with $\Psi_{j}\in\mathfrak{G}_{j}$ ($j=0,1$) and define

[TABLE]

Through this careful definition of distance, we will be able to recover Theorem 1.2 from the following result.

Theorem 2.1**.**

Let $d\geq 1$ and $K\subset(1,2)^{3}$ be a compact subset of admissible triples of exponents. Then, there exists $c>0$ such that for all $\mathbf{p}\in K,\mathbf{f}\in L^{\mathbf{p}}(\mathbb{R}^{2d+1})$, $(2d)\times(2d)$ matrices $A$, and $b\in\mathbb{R}$,

[TABLE]

Proof of Theorem 2.1 $\Rightarrow$ Theorem 1.2.

Suppose that $|\mathcal{T}_{\mathbb{H}^{d}}(\mathbf{f})|\geq(1-\delta)\mathbf{A}_{\mathbf{p}}^{2d+1}\prod_{j}\|f_{j}\|_{p_{j}}$. Since $\mathcal{T}_{\mathbb{H}^{d}}(\mathbf{f})=\mathcal{T}(\mathbf{f},Id,0)$, by Theorem 2.1, we have

[TABLE]

for a constant $C$ independent of $\mathbf{f}$.

By the definitions of $\tilde{\mathcal{O}}(\mathbf{f},Id,0)$ and the distance function, there exist $\Psi\in\mathfrak{G}_{1}$, $b\in\mathbb{R}$, and $A\in Gl(2d)$ such that

[TABLE]

where $\|A^{T}JA\|=O(\delta^{1/2})$ and $b\|A^{T}JA\|=O(\delta^{1/2})$. Observe that since $\Psi$ acts on $(\mathbf{f},Id,0)$, one may view $\Psi$ as an element of $\Sigma(\mathcal{T}_{\mathbb{H}^{d}})$ (and we do so here in interpreting $\Psi f_{j}$). Noting that

[TABLE]

choose $S$ such that equality is attained and write $L=S^{-1}\circ A$.

By (2.3),

[TABLE]

By (2.4), $\|L^{-1}\|^{2}=O(\delta^{1/2})$ and $b\|L^{-1}\|^{2}=O(\delta^{1/2})$, precisely the conclusion of Theorem 1.2 taking $a=1$ in the definition of $\epsilon$-diffuse Gaussian.

∎

To prove Theorem 2.1, we begin by reducing to small perturbations.

Theorem 2.2**.**

Let $d\geq 1$ and $K\subset(1,2)^{3}$ be a compact set of admissible ordered triples of exponents. Then, there exists a function $\delta\mapsto\epsilon(\delta)$ satisfying $\lim_{\delta\to 0}\epsilon(\delta)=0$ with the following property. If $\mathbf{p}\in K$, $\mathbf{f}\in L^{\mathbf{p}}(\mathbb{R}^{2d+1})$, $A$ is a $(2d)\times(2d)$ matrix, and $b\in\mathbb{R}$ such that $|\mathcal{T}(\mathbf{f},A,b)|\geq(1-\delta)\mathbf{A}_{\mathbf{p}}^{2d+1}\prod_{j}\|f_{j}\|_{p_{j}}$, then

[TABLE]

Theorem 2.2 is a qualitative stability result phrased in terms of $\mathcal{T}(\mathbf{f},A,b)$ and the corresponding distance function rather than convolution on $\mathbb{H}^{d}$. We will prove it by translating the following qualitative stability result of Christ [7] into this scheme.

Theorem 2.3**.**

Let $d\geq 1$ and $K\subset(1,2)^{3}$ be a compact set of admissible ordered triples of exponents. Then, there exists a function $\delta\mapsto\epsilon(\delta)$ satisfying $\lim_{\delta\to 0}\epsilon(\delta)=0$ with the following property. Let $\mathbf{p}\in K$, $\mathbf{f}\in L^{\mathbf{p}}(\mathbb{H}^{d})$, and suppose that $\|f_{j}\|_{p_{j}}\neq 0$ for each $j\in\{1,2,3\}$. Let $\delta\in(0,1)$ and suppose that $|\mathcal{T}_{\mathbb{H}^{d}}(\mathbf{f})|\geq(1-\delta)\mathbf{A}_{\mathbf{p}}^{2d+1}\prod_{j}\|f_{j}\|_{p_{j}}$. Then there exists a $\mathbf{p}$-compatible $\epsilon(\delta)$-diffuse ordered triple of Gaussians $\mathbf{G}=(G_{1},G_{2},G_{3})$ such that

[TABLE]

While the version of Theorem 2.3 stated in [7] does not explicitly include the uniformity of $\epsilon(\delta)$ for $\mathbf{p}\in K$, one may easily check this part of the conclusion is satisfied by reviewing the proof.

Proof of Theorem 2.3 $\Rightarrow$ Theorem 2.2.

By a standard approximation argument, it suffices to prove Theorem 2.2 in the case of invertible matrices $A$, as each noninvertible matrix is arbitrarily close to an invertible matrix and $\mathcal{T}(\mathbf{f},A,b)$ is continuous in $A$. (Furthermore, in this scenario, one may choose the distance of the invertible matrix to depend on $\mathbf{f}$.)

Suppose $\Phi(\mathbf{f},A,b)\geq(1-\delta)\mathbf{A}_{\mathbf{p}}^{2d+1}$. Then, applying the symmetries of $\mathcal{T}$, $\Phi(\mathbf{F},Id,0)\geq(1-\delta)\mathbf{A}_{\mathbf{p}}^{2d+1}$, where $\mathbf{F}=e^{-ibt}\mathbf{f}\circ A^{-1}$.

We now write $z=(x,t)$. By Theorem 2.3 (taking $a=1$ via the dilation symmetry), there exist $\Psi\in\Sigma(\mathcal{T}_{\mathbb{H}^{d}}$, $r\in\mathbb{R}$, and $L\in Gl(2d)$ such that

[TABLE]

where $\max\{1,r\}\cdot\|L^{-1}\|^{2}<\epsilon(\delta)$. By (2.5),

[TABLE]

Let $S_{0}$ denote the composition of symplectic matrices found in the symmetries which generate $\Psi$; that is, the matrix $S$ such that for any $(\mathbf{h},M,s)$, $\Psi(\mathbf{h},M,s)=(\tilde{\mathbf{h}},SM,\tilde{s})$ for some $\tilde{\mathbf{h}}\in L^{\mathbf{p}}$ and $\tilde{s}\in\mathbb{R}$. Then, $(e^{-irt}\Psi^{-1}F_{j}\circ L^{-1},S_{0}^{-1}\circ L^{-1},-r)\in\tilde{\mathcal{O}}(\mathbf{f},A,b)$. Since $\Psi$ acts in the case where the attached parameters are $Id$ and [math], we may rightfully view it as an element of $\mathfrak{G}_{1}$.

We now see that

[TABLE]

(In the above, we implicitly used the fact that (2.6) implies $\|e^{-irt}(\Psi^{-1}F_{j})\circ L^{-1}\|_{p_{j}}$ is comparable to $\|g_{j}\|_{p_{j}}$.)

∎

3. Expansion in Group Structure

By the translation scheme developed in Section 2, it suffices to prove Theorem 2.1, and by Theorem 2.2 it suffices to prove it under the assumption of small perturbations.

From here on, let $\mathbf{f}\in L^{\mathbf{p}}(\mathbb{R}^{2d+1})$ denote small, perturbative terms. For this section and the next, fix $b\in\mathbb{R}$ and a $(2d)\times(2d)$ matrix $A$. For $\mathbf{h}\in L^{\mathbf{p}}$, define $\|\mathbf{h}\|_{\mathbf{p}}:=\max_{j}\|h_{j}\|_{p_{j}}$. Our main object of interest is $\Phi(\mathbf{g}+\mathbf{f},A,b)$. We write

[TABLE]

and analyze each of the terms in the expansion in this and the following two sections. Control of the $\frac{\mathcal{T}(\mathbf{g}+\mathbf{f},0,0)}{\prod_{j}\|g_{j}+f_{j}\|_{p_{j}}}$ term will follow partially from the analysis of [6] and will be addressed in Section 6. Control of the other two terms on the right hand side will follow from a trilinear expansion in the function imputs and analysis similar to that of [9].

Upon performing this expansion, the third-order terms will behave in a mildly unexpected manner. Specifically, they will be shown to not be $O(\|\mathbf{f}\|_{\mathbf{p}}^{2}\|A^{T}JA\|)$ through an example which involves the two $f_{j}$ moving out to infinity in opposite directions while minimizing the amount of cancellation. For this reason, it will be helpful to split the $f_{j}$ into pieces near to and far from the origin. The near terms will be analyzed immediately, while the far terms will be addressed later.

As in [5] and [6], let $\eta>0$ be a small parameter to be chosen later (see Theorem 5.1). For each $1\leq j\leq 3$, decompose $f_{j}=f_{j,\sharp}+f_{j,\flat}$, where

[TABLE]

and $f_{j,\flat}=f_{j}-f_{j,\sharp}$.

The main result of this section is the following:

Proposition 3.1**.**

Let $d\geq 1$ and $\mathbf{p}$ be an admissible triple of exponents. Then, there exists constant $C>0$ such that

[TABLE]

The $o(\cdot)$ term will be deemed negligible by the reduction to small perturbations and the $O(\cdot)$ term will be counteracted by a negative term from our treatment of $\frac{\mathcal{T}(\mathbf{g}+\mathbf{f},0,0)}{\prod_{j}\|g_{j}+f_{j}\|_{p_{j}}}$.

We begin by using the trilinearity of $\mathcal{T}$ to expand $\mathcal{T}(\mathbf{g}+\mathbf{f},A,0)-\mathcal{T}(\mathbf{g}+\mathbf{f},0,0)$ into 8 terms of the form

[TABLE]

where the $h_{j}$ are either all $g_{j}$, two $g_{j}$ and one $f_{j}$, one $g_{j}$ and two $f_{j}$, or all $f_{j}$. Since $\mathcal{T}(\cdot,A,0)$ may be written as the integral over the hypersurface $z_{1}\cdot_{A}z_{2}\cdot_{A}z_{3}=0$, the $h_{j}$ are interchangeable. For instance, bounds on $T^{\prime}(f_{1},g_{2},g_{3})$ immediately imply bounds on $T^{\prime}(g_{1},f_{2},g_{3})$ and $T^{\prime}(g_{1},g_{2},f_{3})$; similar implications hold for the case of one $g_{j}$ and two $f_{j}$.

The following two lemmas, proven in [9], will be useful here. By minor abuse of notation, we let $g_{j}(w)=e^{-\gamma_{j}|w|^{2}}$ for $w\in\mathbb{R},w\in\mathbb{R}^{2d}$, or $w\in\mathbb{R}^{2d+1}$.

Lemma 3.2**.**

For all $f\in L^{p_{1}}(\mathbb{R}^{2d})$

[TABLE]

Lemma 3.3**.**

For $\mathbf{g}$ as above,

[TABLE]

The following three lemmas will address the expansion in the current paper.

Lemma 3.4**.**

$T^{\prime}(g_{1},g_{2},g_{3})\leq-C_{d,\mathbf{p}}\|A^{T}JA\|^{2}+O(\|A^{T}JA\|^{3})$**

Proof.

By definition,

[TABLE]

By a simple Taylor expansion, the above is equal to

[TABLE]

where $\alpha:=(t_{1}+t_{2})$ and $\beta:=\sigma(Ax_{1},Ax_{2})$. The higher powers of $\alpha$ and $\beta$ in the Taylor expansion lead to the $O(\|A^{T}JA\|^{3})$ term because $|\sigma(Ax_{1},Ax_{2})|\leq\|A^{T}JA\|\cdot|x_{1}|\cdot|x_{2}|$ and the powers of $x$ and $t$ may be absorbed into the functions $g_{j}$. The resulting sum of integrals converges because the Taylor expansion for the exponential function has summable coefficients. (Formally, one may take the integral over the closed ball of radius $R$ in $\mathbb{R}^{2d}\times\mathbb{R}^{2d}$ so the Taylor expansion converges uniformly. Then, take the limit as $R\to\infty$. This reasoning will also be applied in later lemmas.)

By Lemma 3.2, $\iint g_{1}(z_{1})g_{2}(z_{2})g_{3}(-z_{1}-z_{2})\cdot 2\gamma_{3}\alpha\beta dz_{1}dz_{2}=0$.

By factoring the integral in $z_{j}$ into separate integrals over the $x_{j}$ and $t_{j}$, we see that

[TABLE]

Since $\iint g_{1}(x_{1})g_{2}(x_{2})g_{3}(x_{1}+x_{2})\sigma(Ax_{1},Ax_{2})^{2}dx_{1}dx_{2}=C_{d,\mathbf{p}}\|A^{T}JA\|^{2}$ by Lemma 3.3, it suffices to show that

[TABLE]

Completing the square, we see that

[TABLE]

We now make the change of variables $t_{1}\mapsto t_{1}-\frac{\gamma_{3}}{\gamma_{1}+\gamma_{3}}t_{2},t_{2}\mapsto t_{2}$ so the above becomes

[TABLE]

Since the exponential terms are even in $t_{1}$ and $t_{2}$ and the cross terms $t_{1}t_{2}$ are odd in both variables, this is equal to

[TABLE]

At this point, we use the fact that for $b>0$, $\int_{\mathbb{R}}r^{2}e^{-br^{2}}dr=\frac{1}{2b}\int_{\mathbb{R}}e^{-br^{2}}dr$. Letting $M=\iint e^{-(\gamma_{1}+\gamma_{3})t_{1}^{2}}e^{-\left[\gamma_{2}+\gamma_{3}-\frac{\gamma_{3}^{2}}{\gamma_{1}+\gamma_{3}}\right]t_{2}^{2}}dt_{1}dt_{2}$, the integral in question is equal to

[TABLE]

since $M>0$ and $\gamma_{j}>0$ for all $j$. ∎

Lemma 3.5**.**

$T^{\prime}(f_{1},g_{2},g_{3})=O(\|A^{T}JA\|^{2}\|f_{1}\|_{p_{1}})$.

Proof.

Following the reasoning at the beginning of the proof of Lemma 3.4, we have

[TABLE]

where $\alpha=t_{1}+t_{2}$ and $\beta=\sigma(Ax_{1},Ax_{2})$ as before. The higher-order powers of $\alpha$ and $\beta$ provide a $O(\|A^{T}JA\|^{2}\|f_{1}\|{p_{1}})$ term since $|\sigma(Ax_{1},Ax_{2})|\leq\|A^{T}JA\|\cdot|x_{1}|\cdot|x_{2}|$ and powers may be absorbed into $g_{2}$ and $g_{3}$ to give an $L^{p_{1}^{\prime}}$ function in $z_{1}$. As in the proof of Lemma 3.4, the integrals for all the powers are summable because the original Taylor expansion is summable.

The term coming from $\alpha\beta$ gives 0 by Lemma 3.2. ∎

Naively, one may expect the term $T^{\prime}(f_{1},f_{2},g_{3})$ to be $O(\|A^{T}JA\|\cdot\|\mathbf{f}\|_{\mathbf{p}}^{2})$. This is shown to be false by taking $\phi\in C_{0}^{\infty}(\mathbb{R}^{2d+1})$ to be a bump function near the origin, and letting $f_{1}(z_{1})=\phi(z_{1}-(\lambda,...,\lambda)),f_{2}(z_{2})=\phi(z_{2}+(\lambda,...,\lambda))$ as $|\lambda|\to\infty$. It is here that we will rely heavily on the properties of $f_{j,\sharp}$ in the decomposition $f_{j}=f_{j,\sharp}+f_{j,\flat}$.

Lemma 3.6**.**

$T^{\prime}(f_{1,\sharp},f_{2,\sharp},g_{3})=o(\|\mathbf{f}\|_{\mathbf{p}}^{2}+\|A^{T}JA\|^{2})$* with decay rate depending only on $\eta$.*

In the proof of Lemma 3.6, we will use the following trivial bound

[TABLE]

for arbitrary functions $h_{j}\in L^{p_{j}}$. The proof mimics that of Lemma 3.2 in [9].

Proof.

First, suppose that $\|A^{T}JA\|^{3}\geq\|f_{1,\sharp}\|_{p_{1}}\|f_{2,\sharp}\|_{p_{2}}$. Note that $\|A^{T}JA\|$ may be taken small enough that $\|A^{T}JA\|^{3}\leq\|A^{T}JA\|^{2}$ by our reduction to small perturbations in Theorem 2.2. By (3.3),

[TABLE]

and we are done.

So suppose that $\|A^{T}JA\|^{3}<\|f_{1,\sharp}\|_{p_{1}}\|f_{2,\sharp}\|_{p_{2}}$. Now, for $j=1,2$, write $f_{j,\sharp}=f_{j,\sharp,\leq M_{j}}+f_{j,\sharp,>M_{j}}$, where $f_{j,\sharp,\leq M_{j}}=f_{j,\sharp}{\mathbf{1}}_{B(0,M_{j})}$ and $f_{j,\sharp,>M_{j}}=f_{j,\sharp}{\mathbf{1}}_{B(0,M_{j})^{c}}$. Here, ${\mathbf{1}}_{E}$ refers to the indicator function of the set $E$, $B(z_{0},R)$ refers to the closed ball of radius $R$ centered at $z_{0}$, $E^{c}$ is the complement of the set $E$, and $M_{j}$ is chosen so that

[TABLE]

Note that $M_{j}$ is dependent on $\eta$.

An elementary calculation shows $M_{j}\leq C\log(\|f_{j,\sharp}\|_{p_{j}}^{-1})$. (See the proof of Lemma 3.2 in [9] for details.)

Expand

[TABLE]

The first three of these terms are shown to be $O(\|\mathbf{f}\|_{\mathbf{p}}^{3})$ by combining the trivial bound (3.3) with (3.4).

Let $R=B(0,M_{1})\times B(0,M_{2})\subset\mathbb{R}^{2d}\times\mathbb{R}^{2d}$. Recall by our earlier Taylor expansion that

[TABLE]

In this case, the justification for inclusion of $O(\cdot)$ terms in the integrand is that the integral is over a compact domain; thus, the Taylor expansion converges uniformly.

We see that the absolute value of the integral term containing $-2\gamma_{3}\alpha\beta$ may be controlled by

[TABLE]

The remaining terms may be dealt with similarly, the only difference being that different powers of $\|A^{T}JA\|$ and $M_{j}$ are obtained; however, the end result is always $o(\|\mathbf{f}\|_{\mathbf{p}}^{2})$.

∎

Proof of Proposition 3.1.

Begin by using the trilinearity of $\mathcal{T}$ to expand $\mathcal{T}(\mathbf{g}+\mathbf{f},A,0)-\mathcal{T}(\mathbf{g}+\mathbf{f},0,0)$, expanding again via the decomposition $f_{j}=f_{j,\sharp}+f_{j,\flat}$ when terms contain two $f_{j}$ and one $g_{j}$. Applying Lemmas 3.4, 3.5, and 3.6 to the resulting terms, considering that they apply equally after permutation of indices. The term with three $f_{j}$’s is trivially $O(\|\mathbf{f}\|_{\mathbf{p}}^{3})$.

Note that we may ignore the division by $\prod_{j}\|g_{j}+f_{j}\|_{p_{j}}$ since for small $\|\mathbf{f}\|_{\mathbf{p}}$, this term is approximately the constant value $\prod_{j}\|g_{j}\|_{p_{j}}$; this only results in minor modifications to the constants in the right hand side of the conclusion.

The remaining terms are of the form $T^{\prime}(f_{1,\sharp},f_{2,\flat},g_{3}),T^{\prime}(f_{1,\flat},f_{2,\flat},g_{3})$, or any of the similar forms obtained by permutations; hence, they may not be addressed by Lemma 3.6. However, they may still be controlled by the trivial bound (3.3), resulting in the $O(\|\mathbf{f}_{\sharp}\|_{\mathbf{p}}\|\mathbf{f}_{\flat}\|_{\mathbf{p}}+\|\mathbf{f}_{\flat}\|_{\mathbf{p}}^{2})$ term. ∎

4. Expansion in Twisting Factor

Fix $b\in\mathbb{R}$ and $(2d)\times(2d)$ matrix $A$. Define

[TABLE]

We analyze the expansion of the difference term

[TABLE]

The main result of this section is the following:

Proposition 4.1**.**

Let $d\geq 1$ and $\mathbf{p}$ be an admissible triple of exponents. Then, there exists $C>0$ such that

[TABLE]

As in the previous section, the trilinearity of $T^{\prime\prime}$ gives us 8 terms, each of which has three $g_{j}$, two $g_{j}$ and one $f_{j}$, one $g_{j}$ and two $f_{j}$, or 3 $f_{j}$.

Lemma 4.2**.**

$T^{\prime\prime}(g_{1},g_{2},g_{3})=-C_{d,\mathbf{p}}b^{2}\|A^{T}JA\|^{2}+O(\|A^{T}JA\|^{3}+b^{3}\|A^{T}JA\|^{3})$.

Proof.

As in the proof of Lemma 3.4, we use a Taylor expansion, obtaining

[TABLE]

Here, and again when powers of $\beta$ and $\alpha$ are used with $O(\cdot)$ notation, by $O(\beta^{2})$ we mean that the remaining powers of $\beta$ in the Taylor expansion are of degree 2 or higher. (Issues of convergence may be addressed as in Section 3.)

Similarly,

[TABLE]

We now plug the product of (4.2) and (4.3) into (4.1) with $h_{j}=g_{j}$. Each product of a $O(\cdot)$ term with another term gives a new term which is $O(\beta^{3}+b^{3}\beta^{3})$ and the resulting integral is $O(\|A^{T}JA\|^{3}+b^{3}\|A^{T}JA\|^{3})$.

Factoring out the $g_{3}(-z_{1}-z_{2})$, the remaining terms are $ib\beta-b^{2}\beta^{2}-2ib\gamma_{3}\alpha\beta^{2}-2b^{2}\gamma_{3}\alpha\beta^{3}$. We first note that any integral with a single power of $\alpha$ must give 0 since

[TABLE]

and the powers of $\beta$ only effect the integral in $x_{1},x_{2}$. Second, the $ib\beta$ term gives 0 by Lemma 3.2. The remaining term gives the integral

[TABLE]

by Lemma 3.3. ∎

Lemma 4.3**.**

$T^{\prime\prime}(f_{1},g_{2},g_{3})=O(\|f_{1}\|_{p_{1}}(\|A^{T}JA\|^{2}+b^{2}\|A^{T}JA\|^{2})).$**

Proof.

By the Taylor expansions given in the proof of Lemma 4.2, it suffices to determine bounds for

[TABLE]

As before, the $ib\beta$ term vanishes by Lemma 3.2. For the $-b^{2}\beta^{2}$ term, we see that

[TABLE]

is $O(\|f_{1}\|_{p_{1}}(b^{2}\|A^{T}JA\|^{2}))$ since $\int x_{1}^{2}x_{2}^{2}g_{2}(z_{2})g_{3}(-z_{1}-z_{2})dz_{2}\in L^{p_{1}^{\prime}}(z_{1})$.

The remaining terms may be dealt with similarly

∎

Lemma 4.4**.**

$T^{\prime\prime}(f_{1,\sharp},f_{2,\sharp},g_{3})=o(\|\mathbf{f}\|_{\mathbf{p}}^{2}+(1+b^{2})\|A^{T}JA\|^{2})$.

Proof.

By the reduction to small perturbations, one may take $\|A^{T}JA\|\leq 1$ so that $\|A^{T}JA\|^{3}\leq\|A^{T}JA\|^{2}$. Also take $b\|A^{T}JA\|\leq 1$ so $b^{3}\|A^{T}JA\|^{3}\leq b^{2}\|A^{T}JA\|^{2}$.

As in the proof of Lemma 3.6, the case of $\|A^{T}JA\|^{3}\geq\|f_{1,\sharp}\|_{p_{1}}\|f_{2,\sharp}\|_{p_{2}}$ is taken care of by the trivial bound

[TABLE]

The case of $b^{3}\|A^{T}JA\|^{3}\geq\|f_{1,\sharp}\|_{p_{1}}\|f_{2,\sharp}\|_{p_{2}}$ may be dealt with similarly.

So suppose $\|A^{T}JA\|^{3}<\|f_{1,\sharp}\|_{p_{1}}\|f_{2,\sharp}\|_{p_{2}}$ and $b^{3}\|A^{T}JA\|^{3}<\|f_{1,\sharp}\|_{p_{1}}\|f_{2,\sharp}\|_{p_{2}}$. Let $M_{j}$ and $R$ be as in the proof of Lemma 3.6. Thus, by the proof of Lemma 3.6, it suffices to bound $|T^{\prime\prime}(f_{1,\sharp,\leq M_{1}},f_{2,\sharp,\leq M_{2}},g_{3})|$.

By Taylor expansion,

[TABLE]

Since $|\beta|\leq\|A^{T}JA\|\cdot|x_{1}|\cdot|x_{2}|$, the integral term coming from $ib\beta$ is controlled by

[TABLE]

The remaining terms may be dealt with similarly, instead obtaining different powers of $\|A^{T}JA\|,M_{1}$, and $M_{2}$, though in each case, one may check that the final result is $o(\|\mathbf{f}\|_{\mathbf{p}}^{2})$ due to the presence of a power of $\|A^{T}JA\|$ and the $\log$ bounds for $M_{j}$. ∎

Proof of Proposition 4.1.

As in the proof of Proposition 3.1, the result nearly follows from the trilinearity of $T^{\prime\prime}$, this time combined with the results of Lemmas 4.2, 4.3, and 4.4. Again, there are terms with $f_{j,\flat}$ terms remaining, though by the trivial bound they result in the $O(\|\mathbf{f}_{\sharp}\|_{\mathbf{p}}\|\mathbf{f}_{\flat}\|_{\mathbf{p}}+\|\mathbf{f}_{\flat}\|_{\mathbf{p}}^{2})$ term. ∎

5. Treating the Euclidean Convolution Term

One may hope to complete the proof of Theorem 1.2 by applying Propositions 3.1 and 4.1 to the expansion found in (3.1), along with Theorem 1.1 to address the $\frac{\mathcal{T}(\mathbf{g}+\mathbf{f},0,0)}{\prod_{j}\|g_{j}+f_{j}\|_{p_{j}}}$ term. However, Theorem 1.1 applies when $\mathbf{f}$ represents the projective distance in (1.3), and in our case, the projective distance (2.2) is used– which might not be comparable.

Rather than repeat the entire analysis of [6], it suffices to extract an intermediate theorem proven implicitly in the paper. To state this theorem requires some more definitions.

For $t>0$ and $n=0,1,2,...,$ let $P_{n}^{(t)}$ denote the real-valued polynomial of degree $n$ with positive leading coefficient and $\|P_{n}^{(t)}e^{-t\pi x^{2}}\|_{L^{2}(\mathbb{R})}=1$ which is orthogonal to $P_{k}^{(t)}e^{-t\pi x^{2}}$ for all $0\leq k<n$.

For $d>1$, $\alpha=(\alpha_{1},...,\alpha_{2d+1})\in\{0,1,2,...\}^{2d+1}$, and $x=(x_{1},...,x_{2d+1})\in\mathbb{R}^{2d+1}$, define

[TABLE]

Lastly, for $1\leq j\leq 3$, let $\tau_{j}=\frac{1}{2}p_{j}p_{j}^{\prime}$.

Theorem 5.1** ([6]).**

Let $\delta_{0}>0$ be sufficiently small. There exists $c,\tilde{c}>0$ and a choice of $\eta>0$ in the $f_{j}=f_{j,\sharp}+f_{j,\flat}$ decomposition such that the following holds. Suppose $\|\mathbf{f}\|_{\mathbf{p}}<\delta_{0}$ and $f_{j}$ satisfy the following orthogonality conditions:

• •

$\langle Re(f_{j}),P_{\alpha}^{(\tau_{j})}g_{j}^{p_{j}-1}\rangle=0$* whenever $\alpha=0$, $|\alpha|=1$ and $j\in\{1,2\}$, or $|\alpha|=2$ and $j=3$.*

• •

$\langle Im(f_{j}),P_{\alpha}^{(\tau_{j})}g_{j}^{p_{j}-1}\rangle=0$* whenever $\alpha=0$ or $|\alpha|=1$ and $j=3$.*

Then,

[TABLE]

While Theorem 5.1 was not stated explicitly in [6], (5.1) is effectively the penultimate line in the proof of Theorem 1.1, which used the orthogonality conditions in clear fashion.

Note that the $-\tilde{c}\sum_{j}\|f_{j,\flat}\|_{p_{j}}^{p_{j}}$ will be useful in canceling out the contribution of the $O(\|\mathbf{f}^{\sharp}\|_{\mathbf{p}}\|\mathbf{f}^{\flat}\|_{\mathbf{p}}+\|\mathbf{f}^{\flat}\|_{\mathbf{p}}^{2})$ term.

The following section will show that one may reduce to the case in which the orthogonality conditions of Theorem 5.1 hold.

6. Balancing Lemma

In this section, we prove a Balancing Lemma which will allow us to replace a given $(\mathbf{f},A,b)$ with a nearby one in its orbit that satisfies the orthogonality conditions of Theorem 5.1.

Lemma 6.1** (Balancing Lemma).**

Let $d\geq 1$ and $\mathbf{p}\in(1,2]^{3}$ with $\sum_{j}p_{j}^{-1}=2$. There exists $\delta_{0}>0$ such that if

[TABLE]

then there exists $(\tilde{\mathbf{F}},\tilde{A},\tilde{b})\in\tilde{\mathcal{O}}(\mathbf{F},A,b)$ such that the orthogonality conditions of Theorem 5.1 are satisfied for $\tilde{\mathbf{F}}$.

Proof.

Suppose (6.1) and choose symmetries $\Psi_{i}\in\mathfrak{G}_{i}$ ($i=1,2$) such that

[TABLE]

where $F_{j}^{\prime}=e^{-ibt}F_{j}\circ A^{-1}$. Our goal is to choose $\tilde{\Psi}_{j}\in\mathfrak{G}_{i}$ such that $\tilde{h}_{j}:=\tilde{\Psi}_{0}\tilde{\Psi}_{1}F_{j}^{\prime}-g_{j}$ satisfies the desired orthogonality conditions.

Define $h_{j}:=\Psi_{0}\Psi_{1}F_{j}^{\prime}-g_{j}$ so that

[TABLE]

by defining $\tilde{\Psi}_{1}=\Psi_{1}^{\prime}\Psi_{1}$ and $\Psi_{0}^{\prime}=\tilde{\Psi}_{0}$. Recall that the precise forms of some of the symmetries in $\mathfrak{G}_{1}$ depend on the values of the attached parameters. Since $\Psi_{1}$ originally acts when these parameters are $Id$ and 0, respectively, the form of $\Psi_{1}$ is fixed as such for the above computation, which merely involves triples of functions and not the attached parameters. For instance, here the translation-modulation symmetry is only used as in (2.1), even if it is composed with other symmetries which would normally alter the attached parameters.

In the above, $\Psi_{0}$ is given, but its action on functions may be represented in the form

[TABLE]

for fixed $\beta\in\mathbb{R}$, $L\in Gl(2d)$, and functions $a:\mathbb{R}^{2d+1}\to\mathbb{C}$.

We have some flexibility in determining $\Psi_{0}^{\prime}$, which will be of the form

[TABLE]

for $\gamma\in\mathbb{R}$ and $M\in Gl(2d)$ to be determined later. (In some sense, we will have $\gamma\sim\beta$ and $M\sim L$.)

Let the action of $\Psi_{1}^{\prime}$ be as follows. Given a triple of functions, we first apply the translation symmetry with parameters $u_{j}=(LU_{j},U_{j}^{\prime}),w_{j}=(LW_{j},W_{j}^{\prime})\in\mathbb{R}^{2d}\times\mathbb{R}$ satisfying $w_{1}=u_{2}^{-1},w_{2}=u_{3}^{-1}$, and $w_{3}=u_{1}^{-1}$, then apply the sheer symmetry with linear map $\varphi\circ L^{-1}:\mathbb{R}^{2d}\to\mathbb{R}$, dilate the functions by a factor of $r\in\mathbb{R}$, compose them with $S\in Sp(2d)$, modulate by a factor of $e^{i\xi\cdot x}$ with $\xi\in\mathbb{R}^{2d}$, and finally scale by factors of $a_{j}\in\mathbb{C}$. Thus,

[TABLE]

In combining the above symmetries, we have

[TABLE]

Here, we use $g_{j}(y)=e^{-\gamma_{j}|y|^{2}}$ for $y$ lying in any of $\mathbb{R}$, $\mathbb{R}^{2d}$, or $\mathbb{R}^{2d+1}$ depending on context.

Similarly,

[TABLE]

Write $a_{j}=1+b_{j},M=S^{-1}L(Id+K),r=1+s,\gamma=\beta(r^{2}+U_{j}^{\prime}+W_{j}^{\prime})+\alpha$, and $\xi\cdot x=\zeta+\beta\varphi((Id+K)x)$. Expanding the factors of (6.2), we obtain for the fifth factor

[TABLE]

where $O((|U_{j}|+|W_{j}|+\|K\|+|s|)^{2})$ represents the $L^{p_{j}}$ norm of the remainder term.

Applying similar methods to the sixth factor,

[TABLE]

Since the fifth and sixth factors together include a factor of $g_{j}(x,t)$, we are interested in terms which are not insignificant when multiplied by this $g_{j}$. For the first three factors,

[TABLE]

where $O((|\zeta|+|\alpha|)^{2})$ represents the $L^{p_{j}}$ norm of the remainder term when multiplied by the fifth and sixth factors. Similarly, for the fourth factor

[TABLE]

Combining the above factors gives the following expression for (6.2)

[TABLE]

Recall that $\|L^{T}JL\|,\beta\|L^{T}JL\|=O(\delta)$; thus, terms such as $\|L^{T}JL\|^{2}$ or $\beta\|L^{T}JL\|\cdot\|K\|$ are considered 2nd order in a manner that will be made precise shortly.

We now test our expression for $\tilde{h}_{j}$ in the inner product with the $P_{\alpha}^{\tau_{j}}g_{j}^{p_{j}-1}$. One may check, using the expression from (6.3), that

[TABLE]

The proof will conclude upon applying the Implicit Function theorem to the map $(b_{j},\zeta,\alpha,U_{j},U_{j}^{\prime},K,s,\varphi)\mapsto(\langle\tilde{h}_{j},P_{\alpha}^{\tau_{j}}g_{j}^{p_{j}-1}\rangle:j\in J)$, where $J$ is the collection of indices mentioned in Theorem 5.1. This will guarantee a small neighborhood of $(\langle h_{j},P_{\alpha}^{\tau_{j}}g_{j}^{p_{j}-1}\rangle:j\in J)$ in which $(\langle\tilde{h}_{j},P_{\alpha}^{\tau_{j}}g_{j}^{p_{j}-1}\rangle:j\in J)$ may obtain any value for some set of parameters $b_{j}$,…,etc. The quantitative bounds in the $O(\cdot)$ expressions guarantee that for small enough $\delta_{0}$, the origin is included in this neighborhood. The uniformity in these bounds guarantees the same $\delta_{0}$ works in all cases.

To verify the hypotheses of the Implicit Function theorem, we must show that the map

[TABLE]

is surjective.

One may see by inspection that since $b_{j}\in\mathbb{C}$, the terms containing $b_{j}$ correspond perfectly to the $|\alpha|=0$ cases in the orthogonality conditions. The conditions on $u_{j}$ and $w_{j}$ (and therefore on $(U_{j},U_{j}^{\prime})$ and $(W_{j},W_{j}^{\prime})$) give enough freedom to determine the $|\alpha|=1$ conditions of just the real parts for $j=1,2$. Together, $\xi$ and $\alpha$ give the condition for imaginary parts when $|\alpha|=1$ and $j=3$. Lastly, $K,s$, and $\varphi$ are together in bijective correspondence with the set of symmetric matrices on $\mathbb{R}^{2d}\times\mathbb{R}$; hence, they give the condition for $|\alpha|=2,j=3$.

∎

7. Putting it all Together

Proof of Theorem 1.2.

Recall that by the translation scheme of Section 2, it suffices to prove Theorem 2.1. By Theorem 2.3, it suffices to prove Theorem 2.1 under the assumption of small perturbations.

Let $h_{j}\in L^{p_{j}}(\mathbb{R}^{2d+1})$, $B$ be a $(2d)\times(2d)$ matrix, and $r\in\mathbb{R}$ such that

[TABLE]

By the Balancing Lemma, there exists $F_{j}\in L^{p_{j}}$, a $(2d)\times(2d)$ matrix $A$, and $b\in\mathbb{R}$ such that $(\mathbf{F},A,b)\in\tilde{\mathcal{O}}(\mathbf{h},B,r)$ and the orthogonality conditions of Theorem 5.1 hold for $\mathbf{F}$. Choose $\eta>0$ such that the conclusion of Theorem 5.1 holds. Define $f_{j}=F_{j}-g_{j}$.

Since

[TABLE]

it suffices to show there exists $c>0$ depending only on $d$ and $\mathbf{p}$ such that

[TABLE]

By the expansion

[TABLE]

Propositions 3.1 and 4.1, and Theorem 5.1, we have

[TABLE]

In the above, some sacrifice is made in the constant $c$ to absorb the $o(\cdot)$ terms. Since $p_{j}<2$ for all $j$ and we are working under the assumption of small perturbations, one may absorb the $O(\|\mathbf{f}_{\flat}\|_{\mathbf{p}}^{2})$ term, again at the small expense of constants.

If $\sum_{j}\|f_{j,\flat}\|_{p_{j}}^{p_{j}}$ is small relative to $\|\mathbf{f}\|_{\mathbf{p}}^{2}$, then the $O(\|\mathbf{f}_{\sharp}\|_{\mathbf{p}}\|\mathbf{f}_{\flat}\|_{\mathbf{p}}$ term is negligible, as each $\|f_{j,\flat}\|_{p_{j}}$ is small. (Specifically, one may split into cases where $\|f_{j,\flat}\|_{p_{j}}\geq\|f_{j}\|_{p_{j}}^{(4-p_{j})/2}$ for at least one $j$ or none of the $j$.) However, if $\sum_{j}\|f_{j,\flat}\|_{p_{j}}^{p_{j}}$ is large relative to $\|\mathbf{f}\|_{\mathbf{p}}^{2}$, then the last term dominates (as $p_{j}<2$), and the above is still negligible.

Thus, we are left with (7.1), completing the proof. ∎

8. Variant Theorem Statements

While the main result of [6] is Theorem 1.1, the paper also addresses cases when the hypothesis that $p_{j}<2$ for $1\leq j\leq 3$ is violated. We are able to extend two of these results– one positive, the other negative– to the case of the Heisenberg group without significant further effort.

Theorem 8.1**.**

Let $d\geq 1$ and $\mathbf{p}\in(1,2]^{3}$ be admissible. Then, there exists a $c>0$ with the following property. Let $\mathbf{f}\in L^{\mathbf{p}}(\mathbb{H}^{d})$ and $\|f_{j}\|_{p_{j}}\neq 0$ for each $j\in\{1,2,3\}$. Let $\delta\in(0,1)$ and suppose that $|\mathcal{T}_{\mathbb{H}^{d}}(\mathbf{f})|\geq(1-\delta)\mathbf{A}_{\mathbf{p}}^{2d+1}\prod_{j}\|f_{j}\|_{p_{j}}$. Then there exists a $\mathbf{p}$-compatible $c\sqrt{\delta}$-diffuse ordered triple of Gaussians $\mathbf{G}=(G_{1},G_{2},G_{3})$ such that

[TABLE]

Note Theorem 8.1 is exactly the same as Theorem 1.2, except that the case where one of the $p_{j}=2$ is included at the expense of uniformity in $c$.

Proof.

Without loss of generality, assume $p_{1}=2$. Since $p_{2},p_{3}>1$ and $\sum_{j=1}^{3}p_{j}^{-1}=2$, $p_{2},p_{3}<2$. The proof of Theorem 8.1 mimics that of Theorem 1.2, except one takes $f_{1,\sharp}=f_{1}$, hence $f_{1,\flat}=0$. We must now check every step where the particular properties of $f_{1,\sharp}$ and $f_{1,\flat}$ are used.

In treating the $T^{\prime}(g_{1},f_{2,\sharp},f_{3,\sharp})$ term, the earlier proof of Lemma 3.6 suffices as $f_{1}$ is not present. However, the proof must be modified when treating terms like $T^{\prime}(f_{1,\sharp},f_{2,\sharp},g_{3})$. Only split $f_{2,\sharp}=f_{2,\sharp,>M_{2}}+f_{2,\sharp,\leq M_{2}}$, expanding

[TABLE]

The previous analysis using the trivial bound suffices to control $T^{\prime}(f_{1},f_{2,\sharp,>M_{2}},g_{3})$ as $\|f_{2,\sharp,>M_{2}}\|_{p_{2}}=\|f_{2,\sharp}\|_{p_{2}}^{2}$ still holds. For the remaining term, split $f_{1}=f_{1,>2M_{2}}+f_{1,\leq 2M_{2}}$, where $f_{1,>2M_{2}}=f_{1}{\mathbf{1}}_{B(0,2M_{2})^{c}}$ and $f_{1,\leq 2M_{2}}=f_{1}{\mathbf{1}}_{B(0,2M_{2})}$.

For $T^{\prime}(f_{1,\leq 2M_{2}},f_{2,\sharp,\leq M_{2}},g_{3})$, again the previous analysis will do; this time we simply gain extra powers of $\log(\|f_{2,\sharp}\|_{p_{2}})$ rather than $\log(\|f_{1,\sharp}\|_{p_{1}})$.

For $T^{\prime}(f_{1,>2M_{2}},f_{2,\sharp,\leq M_{2}},g_{3})$, we obtain the following integral:

[TABLE]

As before, we focus on the term with $2\gamma_{3}\alpha\beta$, as bounds for other terms follow similarly.

On the above domain, $|x_{1}+x_{2}|\geq M_{2}$, so $g_{3}(z_{1}+z_{2})\leq e^{-\gamma_{3}(|x_{1}|-M_{2})^{2}-\gamma_{3}(t_{1}+t_{2})^{2}}$. Therefore, the term in question is controlled by

[TABLE]

The proof for Lemma 4.4 is similar.

Lastly, we observe that an equivalent of Theorem 5.1 still holds. In [6], an equivalent of Theorem 8.1 holds for convolution on $\mathbb{R}^{d}$. From the proof of that variant, one may extract a similar intermediate version of Theorem 1.1; however, the $-\tilde{c}\sum_{j=1}^{3}\|f_{j,\flat}\|_{p_{j}}^{p_{j}}$ term is replaced with $-\tilde{c}\sum_{j=2}^{3}\|f_{j,\flat}\|_{p_{j}}^{p_{j}}$. This poses no problems, as $f_{1,\flat}=0$; thus there are no unfavorable terms containing $\|f_{1,\flat}\|_{p_{1}}$ which need be canceled. ∎

The following proposition is found in [6].

Proposition 8.2**.**

Let $\mathbf{p}\in(1,\infty)^{3}$ be admissible and let $d\geq 1$. Suppose that $p_{k}>2$ for some index $1\leq k\leq 3$. Then, there exists no $c>0$ for which

[TABLE]

holds uniformly for all $f_{j}\in L^{p_{j}}(\mathbb{R}^{d})$.

We establish the following variant for our generalized operator.

Proposition 8.3**.**

Let $\mathbf{p}\in(1,\infty)^{3}$ be admissible and let $d\geq 1$. Suppose that $p_{k}>2$ for some index $1\leq k\leq 3$. Then, there exists no $c>0$ for which

[TABLE]

holds uniformly for all $f_{j}\in L^{p_{j}}(\mathbb{R}^{d})$.

Proof.

Suppose for the sake of contradiction that such a $c>0$ exists. Then, taking the case $A=0$ and $b=0$, one recovers the statement

[TABLE]

for all $f_{j}\in L^{p_{j}}(\mathbb{R}^{2d+1})$ uniformly. This is because the symmetries for convolution on $\mathbb{R}^{2d+1}$ contain the symmetries for the generalized operator $\mathcal{T}(\mathbf{f},A,b)$ when $A=0$ and $b=0$. (More symmetries means smaller distance and therefore, weaker statement.) This contradicts Proposition 8.2. Therefore, no such $c>0$ exists. ∎

As a result of Proposition 8.3 and the translation scheme developed in Section 2, Theorem 1.2 does not hold in the case $p_{k}>2$ for some $k$.