# The metaplectic action on modulation spaces

Hartmut Führ *<EMAIL_ADDRESS>and  Irina Shafkulovska †,‡
<EMAIL_ADDRESS>*Lehrstuhl für Geometrie und Analysis, RWTH
Aachen University, D-52056 Aachen, Germany  †Faculty of Mathematics,
University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria
‡Acoustics Research Institute, Austrian Academy of Sciences, Wohllebengasse
12-14, 1040 Vienna, Austria

###### Abstract.

We study the mapping properties of metaplectic operators
$\widehat{S}\in\mathrm{Mp}(2d,\mathbb{\mathbb{R}})$ on modulation spaces of
the type $\mathrm{M}^{p,q}_{m}(\mathbb{\mathbb{R}}^{d})$. Our main result is a
full characterisation of the pairs
$(\widehat{S},\mathrm{M}^{p,q}{}(\mathbb{R}^{d}))$ for which the operator
$\widehat{S}:\mathrm{M}^{p,q}{}(\mathbb{\mathbb{R}}^{d})\to\mathrm{M}^{p,q}{}(\mathbb{\mathbb{R}}^{d})$
is (i) well-defined, (ii) bounded. It turns out that these two properties are
equivalent, and they entail that $\widehat{S}$ is a Banach space automorphism.
For polynomially bounded weight functions, we provide a simple sufficient
criterion to determine whether the well-definedness (boundedness) of
${\widehat{S}:\mathrm{M}^{p,q}{}(\mathbb{\mathbb{R}}^{d})\to\mathrm{M}^{p,q}(\mathbb{\mathbb{R}}^{d})}$
transfers to
$\widehat{S}:\mathrm{M}^{p,q}_{m}(\mathbb{\mathbb{R}}^{d})\to\mathrm{M}^{p,q}_{m}(\mathbb{\mathbb{R}}^{d})$.

Keywords. Modulation spaces, metaplectic operators, symplectic group

AMS subject classifications. 42B35, 46F12

I. Shafkulovska was supported by the Austrian Science Fund (FWF) project
P33217 and thanks the Austrian Academy of Sciences for hosting an internship
at the Acoustics Research Institute.

##  1\. Introduction

This paper studies the interaction of two fundamental objects (or classes of
objects) in time-frequency analysis: the metaplectic group on the one hand,
and the scale of modulation spaces $\mathrm{M}^{p,q}_{m}(\mathbb{R}^{d})$ on
the other. The metaplectic group can be understood as the fundamental symmetry
group in time-frequency analysis; see [10, 15] for more details concerning the
metaplectic group and its role in time-frequency analysis and the
representation theory of the Heisenberg group. The group is often employed to
reduce the study of general cases to more specific, concrete settings; recent
applications of this principle can be found in [13, 14, 18] as recent
illustrations of this principle. Modulation spaces, on the other hand, are
spaces consisting of signals (distributions) whose windowed Fourier transform
exhibits a certain decay, as quantified by a weighted mixed $L^{p}$-norm. We
refer to [5, 15] for more background on modulation spaces.

We are interested in invariance properties, asking under which conditions an
operator $\widehat{S}$ belonging to the metaplectic group, initially defined
as a unitary operator on $L^{2}(\mathbb{R}^{d})$, extends to a bounded
operator
$\mathrm{M}^{p,q}_{m}(\mathbb{R}^{d})\to\mathrm{M}^{p,q}_{m}(\mathbb{R}^{d})$.
The question was first raised in [4], where the invariance of
$\mathrm{M}^{p}(\mathbb{R}^{d})$ under arbitrary metaplectic operators was
instrumental in establishing generalized Strichartz estimates. Given the
relevance both of metaplectic operators and modulation spaces to the field of
time-frequency analysis, understanding their interplay can be regarded as a
fundamental question of independent interest. As the subsequent results and
proofs show, the question turns out to be rather more nuanced than what the
already understood special case $p=q$ and $m\equiv 1$, considered in [4],
suggests.

###  1.1. Contributions

In this subsection, we provide a brief overview of the main results of the
article. For definitions of the various objects in the following theorems, we
refer to the subsequent sections and the background material contained in the
textbooks [5, 15].

Recall that the metaplectic group is the universal cover of the symplectic
group, i.e., there exists a surjective Lie group homomorphism
$\pi^{\mathrm{Mp}}:\mathrm{Mp}(2d,\mathbb{R})\to\mathrm{Sp}(2d,\mathbb{R})$.
Our main new result, restated below as thm:main_unweighted, is a
characterisation of triples $(p,q,\widehat{S})$ of exponents
$p,\,q\in[1,\infty]$ and metaplectic operators
$\widehat{S}\in\mathrm{Mp}(2d,\mathbb{R})$ such that

(1.1)
$\widehat{S}:\mathrm{M}^{p,q}(\mathbb{R}^{d})\to\mathrm{M}^{p,q}(\mathbb{R}^{d})$

is well-defined.

######  Theorem.

Let $p,q\in[1,\infty]$ and $\widehat{S}\in\mathrm{Mp}(2d,\mathbb{R})$ be
given. The following statements are equivalent:

  1. (1)

$\widehat{S}:\mathrm{M}^{p,q}(\mathbb{R}^{d})\to\mathrm{M}^{p,q}(\mathbb{R}^{d})$
is well-defined.

  2. (2)

$\widehat{S}:\mathrm{M}^{p,q}(\mathbb{R}^{d})\to\mathrm{M}^{p,q}(\mathbb{R}^{d})$
is well-defined and bounded.

  3. (3)

One of the following conditions holds:

    1. (i)

$p=q$, or

    2. (ii)

$p\neq q$ and the projection
$\pi^{\mathrm{Mp}}(\widehat{S})=S\in\mathrm{Sp}(2d,\mathbb{R})$ is an upper
block triangular matrix.

This can be lifted to weighted modulation spaces if the metaplectic operator
is compatible with the weight function (see thm:main_weighted).

######  Theorem.

Let $m$ be a moderate polynomially bounded weight, $p,q\in[1,\infty]$. If
$\widehat{S}\in\mathrm{Mp}(2d,\mathbb{R})$ with projection $\pi^{\tiny
Mp}(\widehat{S})=S$ satisfies $m\asymp m\circ S^{-1}$, then $\widehat{S}$ is a
bounded operator from $\mathrm{M}^{p,q}(\mathbb{R}^{d})$ to
$\mathrm{M}^{p,q}(\mathbb{R}^{d})$ if and only if it is a bounded operator
from $\mathrm{M}^{p,q}_{m}(\mathbb{R}^{d})$ to
$\mathrm{M}^{p,q}_{m}(\mathbb{R}^{d})$.

###  1.2. Outline

The paper is structured as follows. The second section covers all necessary
notions related to modulation spaces and metaplectic operators. The third
section contains the proof of the classification for unweighted modulation
spaces. The last section is dedicated to lifting the result from the
unweighted to the weighted setting. Two appendices contain somewhat lengthy
computations of certain ambiguity functions and their dilates.

##  2\. Basic definitions and notation

We begin with a short introduction to time-frequency analysis. The translation
and the modulation operator (time shift and frequency shift) are denoted with
$T_{x}f(t)=f(t-x)$ and $M_{\omega}f(t)=e^{2\pi i\omega\cdot t}f(t)$,
respectively. We will make use of the unitary Fourier transform

(2.1) $\mathcal{F}f(\omega)=\int_{\mathbb{R}^{d}}f(x)e^{-2\pi i\omega\cdot
x}dx,\qquad\omega\in\mathbb{R}^{d},$

pointwise defined on $L^{1}(\mathbb{R}^{d})$ and continuously extended by
duality on $\mathcal{S}^{\prime}(\mathbb{R}^{d})$. Both types of shifts are
unitary representations of the abelian group $\mathbb{R}^{d}$, so they commute
within their class of shifts. However, this does not transfer to their
combinations, as we obtain a non-trivial phase factor

(2.2) $M_{\omega}T_{x}=e^{2\pi i\omega\cdot x}T_{x}M_{\omega},\qquad
x,\omega\in\mathbb{R}^{d}.$

A recurring theme in this paper are intertwining properties. The Fourier
transform can be seen as the elementary example with
$\mathcal{F}T_{x}=M_{-x}\mathcal{F}$,
$\mathcal{F}M_{\omega}=T_{\omega}\mathcal{F}$, which justifies the name
frequency shift for the modulations.

The cross-ambiguity function associated with $f,g\in L^{2}(\mathbb{R}^{d})$ is
the map

(2.3)
$\begin{split}\mathrm{A}(f,g)(x,\omega)&=\int_{\mathbb{R}^{d}}f\left(t+\frac{x}{2}\right)\overline{g\left(t-\frac{x}{2}\right)}e^{-2\pi
i\omega\cdot t}\,dt=\left\langle
f,T_{x/2}M_{\omega}T_{x/2}g\right\rangle.\end{split}$

The latter equality allows to extend the definition of the cross-ambiguity
function for any $f\in\mathcal{S}^{\prime}(\mathbb{R}^{d})$,
$g\in\mathcal{S}(\mathbb{R}^{d})$. It corresponds to the symmetric time-
frequency shift

(2.4)
$\rho(\lambda)=T_{x/2}M_{\omega}T_{x/2}=M_{\omega/2}T_{x}M_{\omega/2},\qquad\lambda=\left(x\,,\,\omega\right)^{t},$

but each parameterization with $\tau\in[0,1]$

(2.5) $\rho_{\tau}(x,\omega)\coloneqq T_{\tau
x}M_{\omega}T_{(1-\tau)x}=e^{2\pi i(1-2\tau)\omega\cdot
x}M_{\tau\omega}T_{x}M_{(1-\tau)\omega}$

induces a sesqui-linear time-frequency representation

(2.6) $\left\langle f,\rho_{\tau}(x,\omega)g\right\rangle=e^{2\pi
i\tau\omega\cdot x-\pi i\omega\cdot x}\left\langle
f,\rho_{1/2}(x,\omega)g\right\rangle=e^{2\pi i\tau\omega\cdot x-\pi
i\omega\cdot x}\mathrm{A}(f,g)(x,\omega).$

Taking $\tau=1/2$ results in the symmetric time-frequency shift. As another
prominent choice, the completely polarized version $\tau=0$ reproduces the
short-time Fourier transform (STFT)

(2.7)
$V_{g}f(x,\omega)=\int_{\mathbb{R}^{d}}f\left(t\right)\overline{g\left(t-x\right)}e^{-2\pi
i\omega\cdot t}\,dt=\left\langle f,M_{\omega}T_{x}g\right\rangle.$

The relation (2.2) does not allow for
$\rho_{\tau}(\lambda+\nu)=\rho_{\tau}(\lambda)\rho_{\tau}(\nu)$ to hold for
arbitrary $\lambda,\,\nu\in\mathbb{R}^{2d}$, but the appearing phase factor
induces a corresponding Heisenberg group $\mathbb{H}_{\tau}$. Topologically,
the Heisenberg group can be identified with
$\mathbb{R}^{d}\times\mathbb{R}^{d}\times\mathbb{R}$. However, this does not
hold on the group level, as the multiplication is non-commuting, chosen in
such a way that $(\lambda,t)\mapsto e^{2\pi it}\rho_{\tau}(\lambda)$ is a
unitary representation on the Hilbert space $L^{2}(\mathbb{R}^{d})$. This
allows for methods from representation theory and gives rise to a non-trivial
family of special intertwining operators, the metaplectic operators, which
will later be addressed in depth. For further exploration of these
parameterized representations, we refer the reader to [2], [3] and [19].

Throughout this paper,
$\left\lVert\mathop{\mathchoice{\vbox{\hbox{\Huge$\cdot$}}}{\vbox{\hbox{\Huge$\cdot$}}}{\vbox{\hbox{\Large$\cdot$}}}{\vbox{\hbox{\small$\cdot$}}}}\right\rVert_{p}\coloneqq\left\lVert\mathop{\mathchoice{\vbox{\hbox{\Huge$\cdot$}}}{\vbox{\hbox{\Huge$\cdot$}}}{\vbox{\hbox{\Large$\cdot$}}}{\vbox{\hbox{\small$\cdot$}}}}\right\rVert_{L^{p}(\mathbb{R}^{d})}$,
denotes the $p$-norm on $L^{p}(\mathbb{R}^{d})$, $p\in[1,\infty]$. We consider
modulation spaces associated to mixed-norm weighted $L^{p}$-spaces, more
formally defined as the spaces $L^{p,q}_{m}(\mathbb{R}^{2d})$ of measurable
functions $f:\mathbb{R}^{d}\times\mathbb{R}^{d}\to\mathbb{C}$ with the norm

$\left\lVert f\right\rVert_{p,q,m}\coloneqq\left\lVert\omega\mapsto\left\lVert
f(\mathop{\mathchoice{\vbox{\hbox{\Huge$\cdot$}}}{\vbox{\hbox{\Huge$\cdot$}}}{\vbox{\hbox{\Large$\cdot$}}}{\vbox{\hbox{\small$\cdot$}}}},\omega)\,m(\mathop{\mathchoice{\vbox{\hbox{\Huge$\cdot$}}}{\vbox{\hbox{\Huge$\cdot$}}}{\vbox{\hbox{\Large$\cdot$}}}{\vbox{\hbox{\small$\cdot$}}}},\omega)\right\rVert_{p}\right\rVert_{q}.$

Here $m:\mathbb{R}^{2d}\to(0,\infty)$ is a weight that is $v$-moderate, i.e.,
there is a $C>0$ with

$m(x+y)\leq C\,v(x)\cdot m(y),\qquad x,y,\in\mathbb{R}^{2d}$

for some weight $v$. Throughout this paper we reserve the term moderate weight
to even weights whose controlling weight can be chosen as
$v(x)=(1+\|x\|)^{N}$, for a suitable exponent $N>0$. Note that this implies
$m(x)\leq C^{\prime}(1+\|x\|)^{N}$, i.e., a moderate weight $m$ is itself
polynomially bounded. The control weight $v$ is submultiplicative, which
amounts to the estimate

(2.8) $v(x+y)\leq C\,v(x)\cdot v(y),\qquad x,y,\in\mathbb{R}^{2d}.$

As a consequence of submultiplicativity, $L^{p,q}_{m}(\mathbb{R}^{2d})$ turns
out to be invariant under translations [15, Proposition 11.1.2.], which is
fundamental for the well-definedness of the associated modulation spaces.

Mixed-norm Lebesgue spaces were systematically explored in [1]. Most
properties, such as completeness, duality and interpolation relations hold in
an analogous fashion to the versions of the standard Lebesgue spaces. The
modulation spaces associated to $L^{p,q}_{m}(\mathbb{R}^{2d})$ are then given
by

$\mathrm{M}^{p,q}_{m}(\mathbb{R}^{d})=\left\\{f\in\mathcal{S}^{\prime}(\mathbb{R}^{d})\mid\mathrm{A}(f,g)\in
L^{p,q}_{m}(\mathbb{R}^{2d})\right\\},$

where $g\in\mathcal{S}(\mathbb{R}^{d})\setminus\\{0\\}$ is fixed. It is well-
known that $\mathrm{M}^{p,q}_{m}(\mathbb{R}^{d})$ is independent of $g$, and
that the associated norm

$\left\lVert
f\right\rVert_{\mathrm{M}^{p,q}_{m}(\mathbb{R}^{d})}\coloneqq\left\lVert\mathrm{A}(f,g)\right\rVert_{p,q,m}$

is independent up to equivalence. Traditionally, the modulation spaces are
defined using the short-time Fourier transform, but the phase factor in (2.6)
has no impact on the definition or the induced norms. Using the ambiguity
function instead of the windowed Fourier transform will make the interaction
with the metaplectic action somewhat easier, specifically in the connection
with twisted convolution in the proof of thm:main_weighted. In the spirit of
the STFT, we will refer to $g\in\mathcal{S}(\mathbb{R}^{d})$ as the window
function of
$\mathrm{A}(\mathop{\mathchoice{\vbox{\hbox{\Huge$\cdot$}}}{\vbox{\hbox{\Huge$\cdot$}}}{\vbox{\hbox{\Large$\cdot$}}}{\vbox{\hbox{\small$\cdot$}}}},g)$.

Modulation spaces are Banach spaces, with natural, continuous embeddings as
closed subspaces of $L^{p,q}_{m}(\mathbb{R}^{2d})$. The reconstruction formula
is given by

(2.9) $\left\langle g,\gamma\right\rangle
f=\mathrm{A}(\mathop{\mathchoice{\vbox{\hbox{\Huge$\cdot$}}}{\vbox{\hbox{\Huge$\cdot$}}}{\vbox{\hbox{\Large$\cdot$}}}{\vbox{\hbox{\small$\cdot$}}}},g)^{*}\,\mathrm{A}(f,\gamma),$

where the adjoint acts (in the weak sense) as

(2.10)
$\mathrm{A}(\mathop{\mathchoice{\vbox{\hbox{\Huge$\cdot$}}}{\vbox{\hbox{\Huge$\cdot$}}}{\vbox{\hbox{\Large$\cdot$}}}{\vbox{\hbox{\small$\cdot$}}}},g)^{*}F=\int_{\mathbb{R}^{2d}}F(x,\omega)\,T_{x/2}M_{\omega}T_{x/2}g\
d(x,\omega).$

The following result was first noted in [9]. See also [5, Section 2.3].

######  Proposition 2.1.

prop:inclusions Let $p,q,r,s\in[1,\infty]$. The following relations hold:

  1. (a)

$\mathrm{M}^{p,q}_{m}(\mathbb{R}^{d})\subseteq\mathrm{M}^{r,s}_{m}(\mathbb{R}^{d})$
if and only if $p\leq r$ and $q\leq s$.

  2. (b)

$(\mathrm{M}^{p,q}_{m}(\mathbb{R}^{d}))^{\prime}=\mathrm{M}^{p^{\prime},q^{\prime}}_{1/m}(\mathbb{R}^{d})$,
if $1\leq p,q<\infty$, with

(2.11) $\left\lvert\left\langle
f,g\right\rangle\right\rvert\lesssim\left\lVert
f\right\rVert_{\mathrm{M}^{p,q}_{m}(\mathbb{R}^{d})}\left\lVert
g\right\rVert_{\mathrm{M}^{p^{\prime},q^{\prime}}_{1/m}(\mathbb{R}^{d})},$

where $p^{\prime}$ and $q^{\prime}$ are the Hölder conjugates of $p$ and $q$,
respectively.

  3. (c)

If the weight function is polynomially bounded,

(2.12)
$\mathcal{S}(\mathbb{R}^{d})\subseteq\mathrm{M}^{1}_{m}(\mathbb{R}^{d})\subseteq\mathrm{M}^{p,q}_{m}(\mathbb{R}^{d})\subseteq\mathrm{M}^{\infty}_{m}(\mathbb{R}^{d})\subseteq\mathcal{S}^{\prime}(\mathbb{R}^{d}).$

For a more detailed overview of modulation spaces, we refer to [5, 6, 11, 15].

###  2.1. The symplectic group and the metaplectic group

Both time and frequency shifts are representations of the abelian group
$\mathbb{R}^{d}$, so they commute within their class, but symmetric time-
frequency shifts do not satisfy $\rho(\lambda)\rho(\nu)=\rho(\lambda+\nu)$ for
arbitrary $\lambda,\nu\in\mathbb{R}^{2d}$.

In order to describe the symplectic group, we first introduce the standard
symplectic form on $\mathbb{R}^{2d}$, which is the bilinear map
$[\cdot,\cdot]:\mathbb{R}^{2d}\times\mathbb{R}^{2d}$ given via

$[(x,\omega),(x^{\prime},\omega^{\prime})]=x^{\prime}\cdot\omega-x\cdot\omega^{\prime}~{},$

where $x,x^{\prime},\omega,\omega^{\prime}\in\mathbb{R}^{d}$. The standard
symplectic matrix

$\mathcal{J}=\begin{pmatrix}0&I_{d}\\\ -I_{d}&0\end{pmatrix},$

with $I_{d}$ denoting the $d$-dimensional identity matrix, allows to express
this bilinear form as
$[\lambda,\lambda^{\prime}]=\lambda\cdot\mathcal{J}\lambda^{\prime}$. Now the
symplectic group $\mathrm{Sp}(2d,\mathbb{R})\leq\mathrm{SL}(2d,\mathbb{R})$
consists of the matrices $S\in\mathrm{GL}(2d,\mathbb{R})$ preserving the
commutator relation, i.e., they satisfy

(2.13)
$[\rho(S\lambda),\rho(S\nu)]_{-}=[\rho(\lambda),\rho(\nu)]_{-},\qquad\lambda,\nu\in\mathbb{R}^{2d},$

where $[\cdot,\cdot]_{-}$ denotes the commutator of linear operators. One
easily verifies that this condition is equivalent to the equation
$S^{T}\mathcal{J}S=\mathcal{J}$. These characterizations readily imply that
the symplectic group is indeed a closed subgroup of the full matrix group
$\mathrm{GL}(2d,\mathbb{R})$.

Due to the block structure of $\mathcal{J}$, symplectic matrices are often
represented in block form as well. The following examples of symplectic
matrices will turn out to be important building blocks of the symplectic
group: let $P,\,Q\in\mathbb{R}^{d\times d}$ be symmetric and
$L\in\mathrm{GL}(d,\mathbb{R})$ be invertible. The following matrices are in
$\mathrm{Sp}(2d,\mathbb{R})$:

(2.14) $U_{P}=\begin{pmatrix}I&P\\\ 0&I\end{pmatrix},\quad
V_{Q}=\begin{pmatrix}I&0\\\ Q&I\end{pmatrix},\quad D_{L}=\begin{pmatrix}L&0\\\
0&L^{-T}\end{pmatrix}.$

The first $d$ elementary symplectic quasi-permutation matrices
$\Pi_{i}\in\mathrm{Sp}(2d,\mathbb{R})$,$\ 1\leq i\leq d$, are given by

(2.15) $\Pi_{i}\ e_{j}=\begin{cases}e_{j}\ ,&\;1\leq j\leq 2d,j\neq i,j\neq
i+d,\\\ -e_{i+d}\ ,&\;j=i,\\\ e_{i}\ ,&\;j=i+d.\end{cases}$

We observe that each $\Pi_{i}$ acts via a combination of a permutation (in
fact, a transposition) and a sign change. Notice that we can represent the
standard symplectic matrix as

(2.16) $\mathcal{J}=\prod\limits_{i=1}^{d}\Pi_{i}.$

For all $1\leq i\leq d$ holds

(2.17) $\Pi_{i+d}\coloneqq\Pi_{i}^{T}=\Pi_{i}^{-1}\;,\quad\Pi_{i+d}\
e_{j}=\begin{cases}e_{j}\ ,&\;1\leq j\leq 2d,j\neq i,j\neq i+d,\\\ e_{i+d}\
,&\;j=i,\\\ -e_{i}\ ,&\;j=i+d.\end{cases}$

Moreover, all $\Pi_{i}$ commute, i.e. $\Pi_{i}\Pi_{j}=\Pi_{j}\Pi_{i}$ for all
$1\leq i,j\leq 2d$. These symplectic matrices are significant not only due to
the clean definition but also for their generating properties.

######  Proposition 2.2 (Factorization in $\mathrm{Sp}(2d,\mathbb{R})$, [7]).

prop:decomposition For all $S\in\mathrm{Sp}(2d,\mathbb{R})$ there is an index
set $J\subseteq\\{1,...,2d\\}$, symmetric matrices $P,Q\in\mathbb{R}^{d\times
d}$ and an invertible matrix $L\in\mathrm{GL}(d,\mathbb{R})$ with

(2.18) $S=\prod\limits_{i\in J}\Pi_{i}\;V_{Q}D_{L}U_{P}.$

The symplectic group can also be understood in terms of the Heisenberg group
$\mathbb{H}$. More precisely, it can be equivalently defined as the group of
linear automorphisms $\mathcal{A}$ on $\left(\mathbb{R}^{2d},+\right)$ which
extend to an automorphism $(\lambda,t)\mapsto(\mathcal{A}t,t)$ of the
Heisenberg group [10]. The property (2.13) and methods from representation
theory, in particular Stone-von Neumann’s theorem, imply the existence of
unitary operators $\widehat{S}$ with

(2.19) $\rho(S\lambda)=\widehat{S}\rho(\lambda)\widehat{S}^{-1},$

that is, $\widehat{S}$ arises as an intertwining operator. They generate a
group $G$ with the composition as the group multiplication. However, any
multiple $\tau\widehat{S}$, $\tau\in\mathbb{T}$, is an intertwining operator,
implying a high redundancy of the group action of $G$ on
$L^{2}(\mathbb{R}^{d})$. Nevertheless, there exists a subgroup of $G$ such
that for all $S\in\mathrm{Sp}(2d,\mathbb{R})$ the said subgroup contains
exactly two corresponding operators $\widehat{S}$. The group is called the
metaplectic group and is denoted by $\mathrm{Mp}(2d,\mathbb{R})$. It is an
explicit realization of the double cover of $\mathrm{Sp}(2d,\mathbb{R})$ and
the projection

$\pi^{\mathrm{Mp}}:\mathrm{Mp}(2d,\mathbb{R})\to\mathrm{Sp}(2d,\mathbb{R})$

is a group homomorphism, with kernel
$\mathrm{ker}(\pi^{\mathrm{Mp}})=\left\\{\mathcal{\mathrm{id},-\mathrm{id}}\right\\}$.
The metaplectic operators are automorphisms on the Schwartz space, and can
therefore be extended by duality to $\mathcal{S}^{\prime}(\mathbb{R}^{d})$
[10, Proposition 4.27]. Standard examples of metaplectic operators are the
unitary dilations
$\mathcal{D}_{L}f(t)=\left\lvert\det{L}\right\rvert^{-1}f(L^{-1}t)$, the
linear chirps $\mathcal{V}_{Q}f(t)=e^{\pi it\cdot Qt}f(t)$ and the Fourier
transform. Indeed,

(2.20)
$\begin{split}\rho(\mathcal{J}\lambda)&=\mathcal{F}\,\rho(\lambda)\,\mathcal{F}^{-1},\\\
\rho(V_{Q}\lambda)&=\mathcal{V}_{Q}\,\rho(\lambda)\,\mathcal{V}_{Q}^{-1},\\\
\rho(D_{L}\lambda)&=\mathcal{D}_{L}\,\rho(\lambda)\,\mathcal{D}_{L}^{-1},\\\
\rho(U_{P}\lambda)&=\left(\mathcal{F}\,\mathcal{V}_{-P}\,\mathcal{F}^{-1}\right)\rho(\lambda)\left(\mathcal{F}\,\mathcal{V}_{-P}\,\mathcal{F}^{-1}\right)^{-1}.\end{split}$

A slightly more explicit description is based on quadratic Fourier transforms.
Detailed constructions can be found in [15, Chapter 9], [10, Chapter 4] and
[6, Chapter 7]. The connection between the ambiguity function and the
metaplectic operators is best summarized by the symplectic covariance of the
ambiguity function:

(2.21) $\mathrm{A}(f,g)\circ S^{-1}\
(\lambda)=\mathrm{A}\left(\widehat{S}f,\widehat{S}g\right)(\lambda).$

Rewriting this equation slightly as

$A(\widehat{S}f,g)(\lambda)=A(f,\widehat{S}^{-1}(g))\circ S^{-1}(\lambda)$

shows that the effect of $\widehat{S}$ on the ambiguity function amounts to
exchanging the window (which induces a well-understood equivalence of
modulation space norms, since the window class is invariant under the
metaplectic action), and the composition operator obtained by letting $S^{-1}$
act on the input variables of the ambiguity function. Hence the focus of the
following section will be on the latter.

##  3\. The metaplectic action on $L^{p,q}(\mathbb{R}^{2d})$ and on
$\mathrm{M}^{p,q}(\mathbb{R}^{d})$

In this section we fully characterize the metaplectic operators leaving
invariant the unweighted modulation spaces $\mathrm{M}^{p,q}(\mathbb{R}^{d})$.
This problem turns out to be rather closely related to a purely measure-
theoretic question, namely invariance of the associated mixed unweighted
spaces $L^{p,q}(\mathbb{R}^{2d})$ under symplectic changes of variable. Before
we explain this aspect of our strategy in more detail, we introduce some
notation concerning dilation operators and their domains and make some
fundamental observations concerning the boundedness of such operators.

######  Proposition 3.1.

prop:closed_operator Let $S\in\mathbb{R}^{2d\times 2d}$ be an invertible
matrix and $V\subseteq L^{p,q}(\mathbb{R}^{2d})$ a closed subspace. Assume

(3.1) $\mathcal{D}_{S}:V\to L^{p,q}(\mathbb{R}^{2d}),\quad f\mapsto f\circ
S^{-1}$

is well-defined. Then $\mathcal{D}_{S}$ is bounded.

###### Proof.

$\mathcal{D}_{S}$ is obviously a linear operator. We will show that it is also
closed. To that end, let
$(f_{n},\mathcal{D}_{S}f_{n})_{n\in\mathbb{N}}\subseteq V\times
L^{p,q}(\mathbb{R}^{2d})$ be an arbitrary Cauchy sequence. Due to the
completeness of $V\times L^{p,q}(\mathbb{R}^{2d})$, the sequence has a limit
$(f,g)$. The convergence in the product space means

(3.2) $\left\lVert f_{n}-f\right\rVert_{L^{p,q}}\to 0\qquad\text{ and
}\qquad\left\lVert\mathcal{D}_{S}f_{n}-g\right\rVert_{L^{p,q}}\to 0.$

By the theorem of Riesz-Fischer, there is a subsequence
$(f_{n_{k}})_{k\in\mathbb{N}}$ with $f_{n_{k}}(x)\to f(x)$ almost everywhere.
Hence we get $\mathcal{D}_{S}f_{n_{k}}(x)\to f\circ S^{-1}(x)$ almost
everywhere. On the other side,

(3.3) $\left\lVert\mathcal{D}_{S}f_{n_{k}}-g\right\rVert_{{p,q}}\to 0$

and we can choose a subsequence $(f_{n_{k_{m}}})_{m\in\mathbb{N}}$ with
$\mathcal{D}_{S}f_{n_{k_{m}}}\to g\ $ almost everywhere. This implies
$g=f\circ S^{-1}$ almost everywhere, which proves the claim. Since both $V$
and $L^{p,q}(\mathbb{R}^{2d})$ are Banach spaces, the closed graph theorem
implies that $\mathcal{D}_{S}$ is bounded. ∎

In the subsequent applications of this result, the closed subspace $V$ will be
either all of $L^{p,q}(\mathbb{R}^{2d})$, or the image space
$A(\cdot,g)(\mathrm{M}^{p,q}(\mathbb{R}^{d}))$ of the modulation space
$\mathrm{M}^{p,q}(\mathbb{R}^{d})$ under the ambiguity operator, with respect
to a fixed nonzero window $g\in S(\mathbb{R}^{d})$. Following standard
terminology, we call $\mathcal{D}_{S}$ everywhere defined on $V$ if

(3.4) $\operatorname{dom}(\mathcal{D}_{S})\coloneqq\\{f\in V\mid f\circ
S^{-1}\in L^{p,q}(\mathbb{R}^{2d})\\}=V.$

With this notation, the interplay between the symplectic action on
$L^{p,q}(\mathbb{R}^{2d})$ and the metaplectic action on
$\mathrm{M}^{p,q}(\mathbb{R}^{d})$ can be best illustrated by the commutative
diagram shown in Figure 1.

${\mathrm{M}^{p,q}(\mathbb{R}^{d})}$${\mathrm{A}(\mathop{\mathchoice{\vbox{\hbox{\Huge$\cdot$}}}{\vbox{\hbox{\Huge$\cdot$}}}{\vbox{\hbox{\Large$\cdot$}}}{\vbox{\hbox{\small$\cdot$}}}},g)(\mathrm{M}^{p,q}(\mathbb{R}^{d}))}$${L^{p,q}(\mathbb{R}^{d})}$${\mathrm{M}^{p,q}(\mathbb{R}^{d})}$${\mathrm{A}(\mathop{\mathchoice{\vbox{\hbox{\Huge$\cdot$}}}{\vbox{\hbox{\Huge$\cdot$}}}{\vbox{\hbox{\Large$\cdot$}}}{\vbox{\hbox{\small$\cdot$}}}},\widehat{S}^{-1}g)(\mathrm{M}^{p,q}(\mathbb{R}^{d}))}$${L^{p,q}(\mathbb{R}^{d})}$$\scriptstyle{\mathrm{A}(\mathop{\mathchoice{\vbox{\hbox{\Huge$\scriptstyle\cdot$}}}{\vbox{\hbox{\Huge$\scriptstyle\cdot$}}}{\vbox{\hbox{\Large$\scriptstyle\cdot$}}}{\vbox{\hbox{\small$\scriptstyle\cdot$}}}},g)}$$\scriptstyle{\widehat{S}}$$\scriptstyle{\mathcal{D}_{S}}$$\scriptstyle{\mathrm{A}(\mathop{\mathchoice{\vbox{\hbox{\Huge$\scriptstyle\cdot$}}}{\vbox{\hbox{\Huge$\scriptstyle\cdot$}}}{\vbox{\hbox{\Large$\scriptstyle\cdot$}}}{\vbox{\hbox{\small$\scriptstyle\cdot$}}}},\widehat{S}^{-1}g)}$
Figure 1. Commutative diagram

We now state the main result of this section.

######  Theorem 3.2.

thm:main_unweighted Let $p,q\in[1,\infty]$ and
$\widehat{S}\in\mathrm{Mp}(2d,\mathbb{R})$ be given. The following statements
are equivalent:

  1. (a)

$\widehat{S}:\mathrm{M}^{p,q}(\mathbb{R}^{d})\to\mathrm{M}^{p,q}(\mathbb{R}^{d})$
is everywhere defined.

  2. (b)

$\widehat{S}:\mathrm{M}^{p,q}(\mathbb{R}^{d})\to\mathrm{M}^{p,q}(\mathbb{R}^{d})$
is everywhere defined and bounded.

  3. (c)

One of the following conditions holds:

    1. (i)

$p=q$, or

    2. (ii)

$p\neq q$ and the projection
$\pi^{\mathrm{Mp}}(\widehat{S})=S\in\mathrm{Sp}(2d,\mathbb{R})$ is an upper
block triangular matrix.

This is a direct consequence of the following theorem.

######  Theorem 3.3.

thm:classification Let $p,q\in[1,\infty]$ and $S\in\mathrm{Sp}(2d,\mathbb{R})$
be given. The following statements are equivalent:

  1. (a)

$\mathcal{D}_{S}:L^{p,q}(\mathbb{R}^{d})\to L^{p,q}(\mathbb{R}^{d})$ is
everywhere defined.

  2. (b)

$\mathcal{D}_{S}:L^{p,q}(\mathbb{R}^{d})\to L^{p,q}(\mathbb{R}^{d})$ is
everywhere defined and bounded.

  3. (c)

$\mathcal{D}_{S}:\mathrm{A}(\mathop{\mathchoice{\vbox{\hbox{\Huge$\cdot$}}}{\vbox{\hbox{\Huge$\cdot$}}}{\vbox{\hbox{\Large$\cdot$}}}{\vbox{\hbox{\small$\cdot$}}}},\widehat{S}^{-1}g)(\mathrm{M}^{p,q}(\mathbb{R}^{d}))\to
L^{p,q}(\mathbb{R}^{d})$ is everywhere defined.

  4. (d)

$\mathcal{D}_{S}:\mathrm{A}(\mathop{\mathchoice{\vbox{\hbox{\Huge$\cdot$}}}{\vbox{\hbox{\Huge$\cdot$}}}{\vbox{\hbox{\Large$\cdot$}}}{\vbox{\hbox{\small$\cdot$}}}},\widehat{S}^{-1}g)(\mathrm{M}^{p,q}(\mathbb{R}^{d}))\to
L^{p,q}(\mathbb{R}^{d})$ is everywhere defined and bounded.

  5. (e)

One of the following conditions holds:

    1. (i)

$p=q$, or

    2. (ii)

$p\neq q$ and $S$ is an upper block triangular matrix.

Let us first comment on the implications that are obvious by now: The
equivalences (a)$\Leftrightarrow$(b) and (c)$\Leftrightarrow$(d) are provided
by Proposition LABEL:prop:closed_operator. The implication (b) $\Rightarrow$
(d) follows from the commutative diagram. Note that the converse (d)
$\Rightarrow$ (b) appears somewhat unexpected at first. Given a densely
defined closed operator $T:Y\to Y$ on a general Banach space, and a closed
subspace $X\subsetneq Y$, there is no reason to expect that boundedness of the
restriction $T|_{X}$ implies boundedness of $T$ itself, and counterexamples to
this expectation are easily constructed. Nonetheless, in the setting studied
here, the implication holds.

Of the remaining implications required to complete the proof of the theorem,
(e) $\Rightarrow$ (a) will turn out to be a fairly standard application of the
linear change of variable formula; see the proof of thm:pq_bounded_unweighted
below. Hence the main challenge in the following will be the proof of the
implication (d) $\Rightarrow$ (e). Here we will rely on the factorization of
arbitrary symplectic matrices provided by prop:decomposition.

But first, we deal with the case $p=q$. Since
$L^{p,p}(\mathbb{R}^{2d})=L^{p}(\mathbb{R}^{2d})$, this case is immediate by
the standard change of variables formula for integrals. The only fact about
symplectic matrices used here is the fact that their determinant equals $1$.

######  Theorem 3.4.

thm:p=q_unweighted The mapping $\mathcal{D}_{S}:L^{p}(\mathbb{R}^{2d})\to
L^{p}(\mathbb{R}^{2d})$ is everywhere defined and norm-preserving for all
$S\in\mathrm{Sp}(2d,\mathbb{R})$ and all $p\in[1,\infty]$.

As a consequence of this observation and the already established implication
(a) $\Rightarrow$ (c), we get the boundedness of
$\widehat{S}:\mathrm{M}^{p}(\mathbb{R}^{d})\to\mathrm{M}^{p}(\mathbb{R}^{d})$,
for any symplectic matrix $S$. This was noted in [4] but was probably known
prior to that.

From now on, let $p,q\in[1,\infty]$ be distinct. In this case, condition (e)
of thm:main_unweighted contains restrictions on the matrix $S$, which arise
from the fact that the definition of the mixed $L^{p,q}$-norm treats the
integration variables $x$ and $\omega$ in an asymmetric manner. Hence any
change of variables that interacts with this asymmetry (”mixes” the $x$\- and
$\omega$-variables) can be expected to be problematic, i.e., potentially lead
to unbounded operators. The challenge in the proof of the remaining direction
lies in making this intuition precise.

The following, somewhat extreme example, based on interchanging the variables
$x,\omega$, gives a hint why the intuition is correct. Pick radially symmetric
functions $f\in L^{p}(\mathbb{R}^{d})\setminus L^{q}(\mathbb{R}^{d})$ and
$g\in L^{q}(\mathbb{R}^{d})\setminus L^{p}(\mathbb{R}^{d})$. The tensor
product $(x,\omega)\mapsto f\otimes g(x,\omega)=f(x)g(\omega)$ is in
$L^{p,q}(\mathbb{R}^{2d})$, but $g\otimes f=\mathcal{D}_{\mathcal{J}}f\otimes
g\notin L^{p,q}(\mathbb{R}^{2d})$, so intuitively, mixing of the variables
should be held under control as much as possible. The example shows that
$\mathcal{D}_{\mathcal{J}}:L^{p,q}(\mathbb{R}^{2d})\to
L^{p,q}(\mathbb{R}^{2d})$ is not well defined.

The following result establishes the implication (e) $\Rightarrow$ (a) of
thm:classification for the case $p\not=q$.

######  Theorem 3.5.

thm:pq_bounded_unweighted Let $A,D\in\mathrm{GL}(d,\mathbb{R})$ and
$B\in\mathbb{R}^{d\times d}$ be given. Denote $S\coloneqq\begin{pmatrix}A&B\\\
0&D\end{pmatrix}$, then

$\mathcal{D}_{S}:L^{p,q}(\mathbb{R}^{2d})\to L^{p,q}(\mathbb{R}^{2d})$

is, up to a constant $C_{S}$, a norm-preserving isomorphism with a bounded
inverse $\mathcal{D}_{S}^{-1}=\mathcal{D}_{S^{-1}}$.

###### Proof.

Let $f\in L^{p,q}(\mathbb{R}^{2d})$ be given.
The inverse of $S$ is

$S^{-1}\coloneqq\begin{pmatrix}A^{-1}&-A^{-1}BD^{-1}\\\
0&D^{-1}\end{pmatrix}.$

Due to the translation invariance of the Lebesgue measure, it holds

(3.5) $\begin{split}\left\lVert\mathcal{D}_{S}f\right\rVert_{{p,q}}{=}&\
\left\lVert\omega\mapsto\left\lVert
f(A^{-1}\mathop{\mathchoice{\vbox{\hbox{\Huge$\cdot$}}}{\vbox{\hbox{\Huge$\cdot$}}}{\vbox{\hbox{\Large$\cdot$}}}{\vbox{\hbox{\small$\cdot$}}}}-A^{-1}BD^{-1}\omega,D^{-1}\omega)\right\rVert_{p}\right\rVert_{q}\\\
=&\ \left\lVert\omega\mapsto\left\lvert\det\
A\right\rvert^{\frac{1}{p}}\left\lVert
f(\mathop{\mathchoice{\vbox{\hbox{\Huge$\cdot$}}}{\vbox{\hbox{\Huge$\cdot$}}}{\vbox{\hbox{\Large$\cdot$}}}{\vbox{\hbox{\small$\cdot$}}}},D^{-1}\omega)\right\rVert_{p}\right\rVert_{q}\\\
=&\ \left\lvert\det\ A\right\rvert^{\frac{1}{p}}\ \left\lvert\det\
D\right\rvert^{\frac{1}{q}}\left\lVert f\right\rVert_{L^{p,q}},\end{split}$

where $a^{\pm\frac{1}{\infty}}\coloneqq 1$.

The inverse of $S$ is also an upper block triangular matrix with
$C_{S^{-1}}=C_{S}^{-1}$. It obviously holds
$\mathcal{D}_{S}\mathcal{D}_{S^{-1}}f=\mathcal{D}_{S^{-1}}\mathcal{D}_{S}f=f$.
Hence $\mathcal{D}_{S}$ is an isomorphism. ∎

Before we finish the proof of our main theorem, we prove a lemma that reduces
the case of general symplectic matrices $S$ to matrices having a specific
structure.

######  Lemma 3.6.

lem:special_symplectic Let $S\in\mathrm{Sp}(2d,\mathbb{R})$ be arbitrary, and
$1\leq p,q\leq\infty$. Then there exists symplectic matrices
$S^{\prime},\,S^{\prime\prime}$ with the following properties:

  1. (a)

There exist $0\leq k\leq d$ and matrices
$Q^{\prime},Q^{\prime\prime}\in\mathbb{R}^{d\times d}$ such that

$S^{\prime}=\prod\limits_{i=k+1}^{d}\Pi_{i}\ V_{Q^{\prime}},\qquad
S^{\prime\prime}=V_{Q^{\prime\prime}}\prod\limits_{i=k+1}^{d}\Pi_{i}~{}.$

  2. (b)

$\mathcal{D}_{S}:\mathrm{A}(\mathop{\mathchoice{\vbox{\hbox{\Huge$\cdot$}}}{\vbox{\hbox{\Huge$\cdot$}}}{\vbox{\hbox{\Large$\cdot$}}}{\vbox{\hbox{\small$\cdot$}}}},\widehat{S}^{-1}g)(\mathrm{M}^{p,q}(\mathbb{R}^{d}))\to
L^{p,q}(\mathbb{R}^{2d})$ is everywhere defined iff $\mathcal{D}_{S^{\prime}}$
or $\mathcal{D}_{S^{\prime\prime}}$ has the same property.

  3. (c)

$S$ is upper block triangular iff $S^{\prime}$ and $S^{\prime\prime}$ are.

###### Proof.

Recall the factorization

$S=\prod\limits_{i\in K}\Pi_{i}\ V_{Q}D_{L}U_{P}~{}$

from prop:decomposition, and note that both $\mathcal{D}_{D_{L}}$ and
$D_{U_{P}}$ are bounded operators with bounded inverses on
$L^{p,q}(\mathbb{R}^{2d})$, by thm:p=q_unweighted. Hence
$\mathcal{D}_{S}:\mathrm{A}(\mathop{\mathchoice{\vbox{\hbox{\Huge$\cdot$}}}{\vbox{\hbox{\Huge$\cdot$}}}{\vbox{\hbox{\Large$\cdot$}}}{\vbox{\hbox{\small$\cdot$}}}},\widehat{S}^{-1}g)(\mathrm{M}^{p,q}(\mathbb{R}^{d}))\to
L^{p,q}(\mathbb{R}^{2d})$ is everywhere defined iff $\mathcal{D}_{M}$ has the
same property, where

$M=\prod\limits_{i\in K}\Pi_{i}\ V_{Q}.$

Furthermore, since the set of upper block triangular matrices is a subgroup
containing $D_{L}$ and $U_{P}$, $M$ is upper block triangular iff $S$ is.

It remains to reduce the case of general index sets $K$ to
$K=\\{k+1,\ldots,d\\}$. To begin with, $\Pi_{i}\Pi_{i+d}=I$, so we can assume
that for no $i\in\\{1,\cdots,d\\}$ both $i$ and $i+d$ are in $I$. To separate
the indices, we define

(3.6) $I_{<}=I\cap\\{1,...,d\\}\quad\text{and}\quad
I_{>}=\\{i\in\\{1,\cdots,d\\}\mid i+d\in I\\}.$

By assumption, $I_{<}$ and $I_{>}$ are disjoint. We define the diagonal matrix
$K_{I}\in\mathbb{R}^{d\times d}$, which serves to correct the sign:

(3.7) $\left(K_{I}\right)_{j,j}=\begin{cases}-1&,\text{ if }\;j\in I_{>},\\\
1&,\text{ otherwise.}\end{cases}$

This results in

(3.8) $\prod\limits_{i\in I}\Pi_{i}=\prod\limits_{i\in
I_{>}}\Pi_{i}^{T}\prod\limits_{i\in
I_{<}}\Pi_{i}=\operatorname{diag}(K_{I},K_{I})\ \prod\limits_{i\in
I_{>}\bigcup I_{<}}\Pi_{i}.$

Finally, we observe

(3.9) $\prod\limits_{i\in I}\Pi_{i}\ e_{j}\neq e_{j}\qquad\text{ if and only
if}\qquad\prod\limits_{i\in I}\Pi_{i}\ e_{j+d}\neq e_{j+d}.$

So as not to have to deal with the order of the indices, we write

(3.10) $\prod\limits_{i\in I_{>}\bigcup
I_{<}}\Pi_{i}=\operatorname{diag}(R_{I},R_{I})\prod\limits_{i=k+1}^{d}\Pi_{i},$

where $k=d-\left\lvert I\right\rvert$ and $R_{I}$ is the permutation matrix
which represents the permutation of the indices in $I$ with
$\\{k+1,\cdots,d\\}$. Altogether, we get the slightly more complicated
decomposition

(3.11) $\begin{split}M&=\operatorname{diag}(K_{I},K_{I})\
\operatorname{diag}(R_{I},R_{I})\prod\limits_{i=k+1}^{d}\Pi_{i}\ V_{Q}\\\
&=\operatorname{diag}(K_{I}R_{I},K_{I}R_{I})\
\underbrace{\prod\limits_{i=k+1}^{d}\Pi_{i}\ V_{Q}}_{\eqqcolon
S^{\prime}}.\end{split}$

Hence, one more appeal to thm:pq_bounded_unweighted provides that
$\mathcal{D}_{S^{\prime}}$ is everywhere defined iff the same holds for
$\mathcal{D}_{S}$. In addition, $S^{\prime}$ is upper block triangular iff $S$
has the same property. Hence (a) and (b) are established for the factorization

$S^{\prime}=\prod\limits_{i=k+1}^{n}\Pi_{i}\ V_{Q}~{}.$

For the verification of (c), observe that the equation

(3.12) $\widetilde{S}=\operatorname{diag}(K_{I}R_{I},K_{I}R_{I})S^{\prime}$

implies that $S^{\prime}$ is upper block triangular iff $\widetilde{S}$ is,
and we already established that this holds iff $S$ is upper block triangular.

We have thus shown the desired statements for the first factorization in (a).
The second factorization, with the associated properties, is obtained by
applying the first factorization to the inverse matrix $S^{-1}$, and taking
inverses. Denoting by $I_{l}$ the indentity matrix in $\mathbb{R}^{l}$, we can
express the inverse of $S^{\prime}$ as

(3.13)
$(S^{\prime})^{-1}=V_{-Q}\prod\limits_{i=k+1}^{d}\Pi_{i}^{T}=\underbrace{V_{-Q}\prod\limits_{i=k+1}^{d}\Pi_{i}}_{\eqqcolon
S^{\prime\prime}}\ \operatorname{diag}(I_{k},-I_{d-k},I_{k},-I_{d-k}).$

The matrix $S^{\prime\prime}$ satisfies the claims by analogous arguments. ∎

We can now finish the proof of our main theorem.

###### Proof of thm:classification.

It remains to prove the implication (d) $\Rightarrow$ (e) of the theorem, for
the case $p\not=q$. Assume that $S\in\mathrm{Sp}(2d,\mathbb{R})$ is not upper
triangular. We intend to establish that
$\mathcal{D}_{S}:\mathrm{A}(\mathrm{M}^{p,q}(\mathbb{R}^{d}),g)\to
L^{p,q}(\mathbb{R}^{2d})$ is unbounded. Recall from the previous lemma that,
without loss of generality, we can assume

(3.14) $S^{-1}=V_{Q}\prod\limits_{i=k+1}^{d}\Pi_{i}\qquad\text{or}\qquad
S^{-1}=\prod\limits_{i=k+1}^{d}\Pi_{i}\ V_{Q}.$

The first factorization will be used to treat the case $p<q$, and the second
one for the case $p>q$. If $Q=0$, this choice is irrelevant.

Now let $\varepsilon>1$, set $E\coloneqq\epsilon I\in\mathbb{R}^{d\times d}$
and denote with $\Delta\coloneqq(1-\varepsilon^{-2})^{\frac{1}{2}}I$. The
matrices

(3.15) $E,\quad
E^{-1},\quad(\varepsilon^{2}-1)^{\scriptscriptstyle\frac{1}{2}}I,\quad\Delta$

are all symmetric positive definite, with $\Delta^{2}+E^{-2}=I$.

It holds

(3.16) $\mathrm{A}\left(\ g\circ(\varepsilon^{2}-1)^{\frac{1}{2}}I,g\
\right)(x,\omega)=\left\lvert\det\ E\right\rvert^{-1}e^{\pi ix\cdot\omega-2\pi
i\omega\cdot E^{-2}x}e^{-\pi x\cdot\Delta^{2}x}e^{-\pi\omega\cdot
E^{-2}\omega}.$

The interested reader can find the detailed computation proving this equation
in Appendix A.

The term $\left\lvert\det\ E\right\rvert^{-1}e^{\pi ix\cdot\omega-2\pi
i\omega\cdot E^{-2}x}$ has no effect on the norm estimate. Therefore we focus
on $f(x,\omega)\coloneqq e^{-\pi x\cdot\Delta^{2}x}e^{-\pi\omega\cdot
E^{-2}\omega}$ and $f\circ S^{-1}$. It holds

(3.17) $\left\lVert f\right\rVert_{{p,q}}=\left\lVert
g\circ\Delta\right\rVert_{p}\left\lVert g\circ
E^{-1}\right\rVert_{q}\overset{\scriptscriptstyle\ref{lemma:NormFormel}}{\asymp}\det(\Delta^{2})^{-\frac{1}{2p}}\det(E^{-2})^{-\frac{1}{2q}}.$

We denote the blocks of $S^{-1}$ by

(3.18) $S^{-1}=:\begin{pmatrix}A&B\\\ C&D\end{pmatrix}$

and compute $f\circ S^{-1}$ as

(3.19) $(f\circ
S^{-1})(x,\omega)=\exp\left(-\pi(Ax+B\omega)\cdot\Delta^{2}(Ax+B\omega)-\pi(Cx+D\omega)\cdot
E^{-2}(Cx+D\omega)\right).$

By expanding and grouping the quadratic terms, one can show that

(3.20) $\left\lVert f\circ S^{-1}\right\rVert_{{p,q}}\asymp\left\lvert\det\
\Sigma\right\rvert^{-\frac{1}{2p}}\left\lvert\det\
\Omega\right\rvert^{-\frac{1}{2q}}$

for all $p,q\in[1,\infty],\ p\neq q$, where

(3.21) $\begin{split}\Sigma&\coloneqq A^{T}\Delta^{2}A+C^{T}E^{-2}C,\\\
\beta&\coloneqq B^{T}\Delta^{2}A+D^{T}E^{-2}C,\\\ \Omega&\coloneqq
B^{T}\Delta^{2}B+D^{T}E^{-2}D-\beta\Sigma^{-1}\beta^{T}.\end{split}$

The technical calculation is provided in Appendix B.

We now compute $\Sigma,\beta$, and $\Omega$ for a few extremal cases and
compare the norms. Throughout, we denote with
$I_{k}=\operatorname{diag}(1,\dots,1,0,\dots,0)\in\mathbb{R}^{d\times d}$ the
diagonal matrix with $k$ ones and $d-k$ zeros.
Case 1: $S^{-1}=\prod\limits_{i=k+1}^{d}\Pi_{i}\
V_{0}=\begin{pmatrix}I_{k}&I-I_{k}\\\ I_{k}-I&I_{k}\end{pmatrix}$.
We keep in mind that $\Delta,\,E\in\mathbb{R}I$, i.e., $\Delta$ and $E$ are
multiples of $I$, as well as the fact that $(I-I_{k})$ and $I_{k}$ are
diagonal matrices with $(I-I_{k})\cdot I_{k}=0$. It thereby holds

(3.22)
$\begin{split}\Sigma&=A^{T}\Delta^{2}A+C^{T}E^{-2}C=I_{k}\Delta^{2}I_{k}+(I_{k}-I)E^{-2}(I_{k}-I)=\Delta^{2}I_{k}+E^{-2}(I-I_{k}),\\\\[3.01385pt]
\beta&=B^{T}\Delta^{2}A+D^{T}E^{-2}C=(I_{k}-I)\Delta^{2}I_{k}+I_{k}E^{-2}(I-I_{k})=0+0=0\\\\[3.01385pt]
&\text{and}\\\\[3.01385pt]
\Omega&=B^{T}\Delta^{2}B+D^{T}E^{-2}D-\beta\Sigma^{-1}\beta^{T}\\\
&=(I-I_{k})\Delta^{2}(I-I_{k})+I_{k}E^{-2}I_{k}=\Delta^{2}(I-I_{k})+E^{-2}I_{k}.\\\\[3.01385pt]
\end{split}$

For the sake of contradiction, assume $\mathcal{D}_{S}$ were bounded. In that
case, it would hold

(3.23) $\begin{split}1&\gtrsim\left\lvert\det\
\Delta^{2}\right\rvert^{\frac{1}{2p}}\ \left\lvert\det\
E^{-2}\right\rvert^{\frac{1}{2q}}\ \left\lvert\det\
\Sigma\right\rvert^{-\frac{1}{2p}}\ \left\lvert\det\
\Omega\right\rvert^{-\frac{1}{2q}}\\\\[3.01385pt]
&=(1-\varepsilon^{-2})^{\frac{d}{2p}}\ (\varepsilon^{-2})^{\frac{d}{2q}}\
(1-\varepsilon^{-2})^{-\frac{k}{2p}}\ (\varepsilon^{-2})^{-\frac{d-k}{2p}}\
(1-\varepsilon^{-2})^{-\frac{d-k}{2q}}\
(\varepsilon^{-2})^{-\frac{k}{2q}}\\\\[3.01385pt]
&=(1-\varepsilon^{-2})^{\frac{d}{2p}-\frac{k}{2p}-\frac{d-k}{2q}}(\varepsilon^{-2})^{\frac{d}{2q}-\frac{d-k}{2p}-\frac{k}{2q}}\\\\[3.01385pt]
&=(1-\varepsilon^{-2})^{(d-k)(\frac{1}{2p}-\frac{1}{2q})}\
(\varepsilon^{-2})^{(d-k)(\frac{1}{2q}-\frac{1}{2p})}\\\\[3.01385pt]
&=\left(\frac{1-\varepsilon^{-2}}{\varepsilon^{-2}}\right)^{(d-k)(\frac{1}{2p}-\frac{1}{2q})}=\left(\varepsilon^{2}-1\right)^{(d-k)(\frac{1}{2p}-\frac{1}{2q})}.\\\\[3.01385pt]
\end{split}$

The matrix is not the identity matrix, so $k<d$. The exponent is not $0$ due
to $p\neq q$. Depending on the sign of the exponent, by observing the
expression as $\varepsilon\to\infty$ or $\varepsilon\searrow 1$, we conclude
that the estimate cannot hold. Looking at $f$, it is clear that the exact
index set of the permutations is irrelevant, only its cardinality matters.
Case 2: $p<q$, $Q\neq 0$ and $S^{-1}=V_{-Q}\ \prod\limits_{i=k+1}^{d}\Pi_{i}.$

Assume there were a $g_{0}\in\mathrm{A}(\mathrm{M}^{p,q}(\mathbb{R}^{d}),g)$
with

(3.24)
$\left\lvert\mathcal{D}_{V_{Q}}\,g_{0}\right\rvert=\left\lvert\mathrm{A}\left(g\circ(\varepsilon^{2}-1)^{\frac{1}{2}}I,\
g\right)\right\rvert.$

In that case, holds

(3.25) $\left\lVert g_{0}\right\rVert_{{p,q}}=\left\lVert\left\lvert\det\
E\right\rvert^{-1}f\circ
V_{Q}\right\rVert_{{p,q}}\quad\text{and}\quad\left\lVert g_{0}\circ
S^{-1}\right\rVert_{{p,q}}=\left\lVert\left\lvert\det\
E\right\rvert^{-1}f\circ\prod\limits_{i=k+1}^{d}\Pi_{i}\right\rVert_{{p,q}}.$

According to Case 1, it holds

(3.26) $\left\lVert g_{0}\circ
S^{-1}\right\rVert_{{p,q}}\asymp\left\lvert\det\
E\right\rvert^{-1}(1-\varepsilon^{-2})^{-\frac{k}{2p}}\
(\varepsilon^{-2})^{-\frac{d-k}{2p}}\ (1-\varepsilon^{-2})^{-\frac{d-k}{2q}}\
(\varepsilon^{-2})^{-\frac{k}{2q}}.$

For $\left\lVert g_{0}\right\rVert_{{p,q}}$, we compute the corresponding
$\Sigma,\ \beta$ and $\Omega$. To this end, we write $Q=U\Lambda U^{T}$, where
$U\in\mathrm{O}(d,\mathbb{R})$ and $\Lambda$ diagonal, and use
$\Delta,\,E\in\mathbb{R}I$ to obtain

(3.27) $\displaystyle\Sigma$
$\displaystyle=A^{T}\Delta^{2}A+C^{T}E^{-2}C=\Delta^{2}+QE^{-2}Q=U\left(I-E^{-2}+\Lambda^{2}E^{-2}\right)U^{T}\
$ (3.28)
$\displaystyle=U\left(\operatorname{diag}\left(1-\varepsilon^{-2}+\Lambda^{2}_{i,i}\
\varepsilon^{-2}\right)_{1\leq i\leq d}\right)U^{T},$ (3.29) hence (3.30)
$\displaystyle\Sigma^{-1}$ $\displaystyle=U\
\operatorname{diag}\left(\frac{1}{1-\varepsilon^{-2}+\Lambda^{2}_{i,i}\
\varepsilon^{-2}}\right)_{1\leq i\leq d}\ U^{T},$ (3.31) $\displaystyle\beta$
$\displaystyle=B^{T}\Delta^{2}A+D^{T}E^{-2}C=E^{-2}Q~{},$ (3.32)
$\displaystyle\Omega$ $\displaystyle=\
B^{T}\Delta^{2}B+D^{T}E^{-2}D-\beta\Sigma^{-1}\beta^{T}=E^{-2}-E^{-2}Q\Sigma^{-1}QE^{-2}=$
(3.33) $\displaystyle=\ E^{-2}\left(I-U\Lambda U^{T}\ U\
\operatorname{diag}\left(\frac{1}{1-\varepsilon^{-2}+\Lambda^{2}_{i,i}\
\varepsilon^{-2}}\right)_{1\leq i\leq d}\ U^{T}\ U\Lambda U^{T}\
E^{-2}\right)$ (3.34) $\displaystyle=\ E^{-2}\ U\left(I-\
\operatorname{diag}\left(\frac{\Lambda^{2}_{i,i}\varepsilon^{-2}}{1-\varepsilon^{-2}+\Lambda^{2}_{i,i}\
\varepsilon^{-2}}\right)_{1\leq i\leq d}\right)\ U^{T}$ (3.35)
$\displaystyle=\ U\ E^{-2}\
\operatorname{diag}\left(\frac{1-\varepsilon^{-2}+\Lambda^{2}_{i,i}\
\varepsilon^{-2}\
-\Lambda^{2}_{i,i}\varepsilon^{-2}}{1-\varepsilon^{-2}+\Lambda^{2}_{i,i}\
\varepsilon^{-2}}\right)_{1\leq i\leq d}\ U^{T}$ (3.36) $\displaystyle=\ U\
E^{-2}\Delta^{2}\
\operatorname{diag}\left(\frac{1}{1-\varepsilon^{-2}+\Lambda^{2}_{i,i}\
\varepsilon^{-2}}\right)_{1\leq i\leq d}\ U^{T}.$

Assume $\mathcal{D}_{S}$ were bounded. Then it would hold

(3.37) $\displaystyle 1$ $\displaystyle\gtrsim\left\lVert
g_{0}\right\rVert^{-1}_{{p,q}}\left\lVert g_{0}\circ
S^{-1}\right\rVert_{{p,q}}\asymp\left\lvert\det\ E\right\rvert\left\lvert\det\
\Sigma\right\rvert^{\frac{1}{2p}}\ \left\lvert\det\
\Omega\right\rvert^{\frac{1}{2q}}\;\left\lVert g_{0}\circ
S^{-1}\right\rVert_{{p,q}}$ (3.38)
$\displaystyle=\left(\prod\limits_{i=1}^{d}\Lambda^{2}_{i,i}\varepsilon^{-2}+(1-\varepsilon^{-2})\right)^{\frac{1}{2p}}\
\left(\prod\limits_{i=1}^{d}\frac{1}{\Lambda^{2}_{i,i}\varepsilon^{-2}+(1-\varepsilon^{-2})}\right)^{\frac{1}{2q}}$
(3.39)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad(1-\varepsilon^{-2})^{\frac{d}{2q}}(\varepsilon^{-2})^{\frac{d}{2q}}\left\lvert\det\
E\right\rvert\cdot\left\lVert\mathcal{D}_{S}g_{0}\right\rVert_{{p,q}}$ (3.40)
$\displaystyle=\left(\prod\limits_{i=1}^{d}\Lambda^{2}_{i,i}\varepsilon^{-2}+(1-\varepsilon^{-2})\right)^{\frac{1}{2p}-\frac{1}{2q}}\
(1-\varepsilon^{-2})^{\frac{d}{2q}}(\varepsilon^{-2})^{\frac{d}{2q}}$ (3.41)
$\displaystyle\qquad\qquad\qquad\qquad(1-\varepsilon^{-2})^{-\frac{k}{2p}}\
(\varepsilon^{-2})^{-\frac{d-k}{2p}}\ (1-\varepsilon^{-2})^{-\frac{d-k}{2q}}\
(\varepsilon^{-2})^{-\frac{k}{2q}}\qquad\qquad$ (3.42)
$\displaystyle=\left(\prod\limits_{i=1}^{d}\Lambda^{2}_{i,i}\varepsilon^{-2}+(1-\varepsilon^{-2})\right)^{\frac{1}{2p}-\frac{1}{2q}}\
(1-\varepsilon^{-2})^{-k(\frac{1}{2p}-\frac{1}{2q})}(\varepsilon^{-2})^{-(d-k)(\frac{1}{2p}-\frac{1}{2q})}$
(3.43) $\displaystyle=\left(\
\prod\limits_{i=1}^{k}\frac{\Lambda^{2}_{i,i}\varepsilon^{-2}+(1-\varepsilon^{-2})}{1-\varepsilon^{-2}}\right)^{\frac{1}{2p}-\frac{1}{2q}}\cdot\quad\left(\
\prod\limits_{i=k+1}^{d}\frac{\Lambda^{2}_{i,i}\varepsilon^{-2}+(1-\varepsilon^{-2})}{\varepsilon^{-2}}\right)^{\frac{1}{2p}-\frac{1}{2q}}$
(3.44)
$\displaystyle=\left(\prod\limits_{i=1}^{k}\frac{\Lambda^{2}_{i,i}}{\varepsilon^{2}-1}+1\right)^{\frac{1}{2p}-\frac{1}{2q}}\;\left(\prod\limits_{i=k+1}^{d}\Lambda^{2}_{i,i}+\varepsilon^{2}-1\right)^{\frac{1}{2p}-\frac{1}{2q}}.$

The exponent $\frac{1}{2p}-\frac{1}{2q}$ is positive. If $k<d$, then the
second product is not empty. The first product is bounded from below by $1$.
Since

(3.45)
$\lim\limits_{\varepsilon\to\infty}\prod\limits_{i=k+1}^{d}\Lambda^{2}_{i,i}+\varepsilon^{2}-1=\infty,$

we meet a contradiction.

If $k=d$, then the second product is empty, i.e., it equals $1$. The symmetric
matrix $Q$ has at least one eigenvalue $\Lambda_{i,i}\neq 0$. In this case,

(3.46) $\lim\limits_{\varepsilon\to
1}\prod\limits_{j=1}^{d}\frac{\Lambda^{2}_{j,j}}{\varepsilon^{2}-1}+1\geq
1^{d-1}\lim\limits_{\varepsilon\to
1}\frac{\Lambda^{2}_{i,i}}{\varepsilon^{2}-1}+1=\infty,$

contradicts the assumption.
Overall, $\mathcal{D}_{S}$ cannot be bounded.
Case 3: $p>q$, $Q\neq 0$ and $S^{-1}=\prod\limits_{i=k+1}^{d}\Pi_{i}^{T}\
V_{Q}.$
Similarly to the previous case, we are looking for
$h_{0}\in\mathrm{A}(\mathop{\mathchoice{\vbox{\hbox{\Huge$\cdot$}}}{\vbox{\hbox{\Huge$\cdot$}}}{\vbox{\hbox{\Large$\cdot$}}}{\vbox{\hbox{\small$\cdot$}}}},g)(\Omega^{p,q}(\mathbb{R}^{d}))$
with

(3.47) $\left\lvert
h_{0}\circ\prod\limits_{i=k+1}^{d}\Pi_{i}\right\rvert=\left\lvert\mathrm{A}\left(\
g\circ(\varepsilon^{2}-1)^{\frac{1}{2}}I,\ g\ \right)\right\rvert.$

Then, it holds

(3.48) $\left\lVert h_{0}\right\rVert_{{p,q}}=\left\lVert\left\lvert\det\
E\right\rvert^{-1}f\circ\prod\limits_{i=k+1}^{d}\Pi_{i}\right\rVert_{{p,q}}\quad\text{and}\quad\left\lVert
h_{0}\circ S^{-1}\right\rVert_{{p,q}}=\left\lVert\left\lvert\det\
E\right\rvert^{-1}f\circ V_{Q}\right\rVert_{{p,q}}.$

Under the assumption that $\mathcal{D}_{S}$ is bounded, the computation in the
second step would imply

(3.49) $\begin{split}1&\gtrsim\left\lVert
h_{0}\right\rVert^{-1}_{{p,q}}\left\lVert\mathcal{D}_{S}\,h_{0}\right\rVert_{{p,q}}\asymp\left(\prod\limits_{i=1}^{k}\frac{\Lambda^{2}_{i,i}}{\varepsilon^{2}-1}+1\right)^{\frac{1}{2q}-\frac{1}{2p}}\;\left(\prod\limits_{i=k+1}^{d}\Lambda^{2}_{i,i}+\varepsilon^{2}-1\right)^{\frac{1}{2q}-\frac{1}{2p}}.\end{split}$

We have already seen that this cannot be true. Therefore, $\mathcal{D}_{S}$ is
unbounded.
Step 4: (The existence of $g_{0}$ and $h_{0}$)
It is known that $\varphi\in\mathcal{S}(\mathbb{R}^{d})$ is equivalent to
$\mathrm{A}(\varphi,\gamma)\in\mathcal{S}(\mathbb{R}^{2d})$ for all
$\gamma\in\mathcal{S}(\mathbb{R}^{d})$, in particular,
$\mathrm{A}(\varphi,\gamma)\in L^{p,q}(\mathbb{R}^{2d})$.
For a given $R\in\mathrm{Sp}(2d,\mathbb{R})$ and
$\varphi\in\mathcal{S}(\mathbb{R}^{d})$, we are looking for a
$\psi\in\mathrm{M}^{p,q}(\mathbb{R}^{d})$ with

(3.50)
$\left\lvert\mathrm{A}(\varphi,g)\right\rvert=\left\lvert\mathcal{D}_{R}\mathrm{A}(\psi,g)\right\rvert=\left\lvert\mathrm{A}(\widehat{R}{\psi},\widehat{R}g)\right\rvert.$

We can actually solve

(3.51) $\mathrm{A}(\widehat{R}{\psi},\widehat{R}g)=\mathrm{A}(\varphi,g)$

for $\psi$. For that, we choose $\gamma\in\mathcal{S}(\mathbb{R}^{d})$ with
$\left\langle\gamma,g\right\rangle\neq
0\neq\left\langle\gamma,\widehat{R}g\right\rangle$. The function $\gamma$
exists because $\mathcal{S}(\mathbb{R}^{d})$ is not two-dimensional. With the
inversion formula, we get

(3.52)
$\mathrm{A}(\mathop{\mathchoice{\vbox{\hbox{\Huge$\cdot$}}}{\vbox{\hbox{\Huge$\cdot$}}}{\vbox{\hbox{\Large$\cdot$}}}{\vbox{\hbox{\small$\cdot$}}}},\gamma)^{*}\mathrm{A}(\mathop{\mathchoice{\vbox{\hbox{\Huge$\cdot$}}}{\vbox{\hbox{\Huge$\cdot$}}}{\vbox{\hbox{\Large$\cdot$}}}{\vbox{\hbox{\small$\cdot$}}}},\widehat{R}g)\widehat{R}\psi=\mathrm{A}(\mathop{\mathchoice{\vbox{\hbox{\Huge$\cdot$}}}{\vbox{\hbox{\Huge$\cdot$}}}{\vbox{\hbox{\Large$\cdot$}}}{\vbox{\hbox{\small$\cdot$}}}},\gamma)^{*}\mathrm{A}(\mathop{\mathchoice{\vbox{\hbox{\Huge$\cdot$}}}{\vbox{\hbox{\Huge$\cdot$}}}{\vbox{\hbox{\Large$\cdot$}}}{\vbox{\hbox{\small$\cdot$}}}},g)\varphi,$

i. e.,

(3.53)
$\left\langle\gamma,\widehat{R}g\right\rangle^{-1}\widehat{R}\psi=\left\langle\gamma,g\right\rangle^{-1}\varphi.$

In our case,
$\varphi=g\circ(\varepsilon^{2}-1)^{\scriptscriptstyle\frac{1}{2}}I$, so the
claim can be applied to $g_{0}$ and $h_{0}$. A possible choice would be

(3.54)
$\psi=\frac{\left\langle\gamma,\widehat{R}g\right\rangle}{\left\langle\gamma,g\right\rangle}\widehat{R}^{-1}\varphi.$

To summarize,

(3.55)
$\mathcal{D}_{S}:\mathrm{A}(\mathop{\mathchoice{\vbox{\hbox{\Huge$\cdot$}}}{\vbox{\hbox{\Huge$\cdot$}}}{\vbox{\hbox{\Large$\cdot$}}}{\vbox{\hbox{\small$\cdot$}}}},\gamma)(\mathrm{M}^{p,q}(\mathbb{R}^{d}))\to
L^{p,q}(\mathbb{R}^{2d})$

cannot be bounded, hence it cannot be everywhere defined on
$\mathrm{A}(\mathop{\mathchoice{\vbox{\hbox{\Huge$\cdot$}}}{\vbox{\hbox{\Huge$\cdot$}}}{\vbox{\hbox{\Large$\cdot$}}}{\vbox{\hbox{\small$\cdot$}}}},\gamma)(\mathrm{M}^{p,q}(\mathbb{R}^{d}))$.
∎

##  4\. Weighted Spaces

###  4.1. The Lifting on $L^{p,q}_{m}(\mathbb{R}^{2d})$

Figure 2 contains the weighted version of the fundamental commutative diagram.

${\mathrm{M}^{p,q}_{m}(\mathbb{R}^{d})}$${\mathrm{A}(\mathop{\mathchoice{\vbox{\hbox{\Huge$\cdot$}}}{\vbox{\hbox{\Huge$\cdot$}}}{\vbox{\hbox{\Large$\cdot$}}}{\vbox{\hbox{\small$\cdot$}}}},g)(\mathrm{M}^{p,q}_{m}(\mathbb{R}^{d}))}$${L^{p,q}_{m}(\mathbb{R}^{2d})}$${\mathrm{M}^{p,q}_{m}(\mathbb{R}^{d})}$${\mathrm{A}(\mathop{\mathchoice{\vbox{\hbox{\Huge$\cdot$}}}{\vbox{\hbox{\Huge$\cdot$}}}{\vbox{\hbox{\Large$\cdot$}}}{\vbox{\hbox{\small$\cdot$}}}},\widehat{S}^{-1}g)(\mathrm{M}^{p,q}_{m}(\mathbb{R}^{d}))}$${L^{p,q}_{m}(\mathbb{R}^{2d})}$$\scriptstyle{\mathrm{A}(\mathop{\mathchoice{\vbox{\hbox{\Huge$\scriptstyle\cdot$}}}{\vbox{\hbox{\Huge$\scriptstyle\cdot$}}}{\vbox{\hbox{\Large$\scriptstyle\cdot$}}}{\vbox{\hbox{\small$\scriptstyle\cdot$}}}},g)}$$\scriptstyle{\widehat{S}}$$\scriptstyle{(\mathcal{D}_{S})_{m}}$$\scriptstyle{\mathrm{A}(\mathop{\mathchoice{\vbox{\hbox{\Huge$\scriptstyle\cdot$}}}{\vbox{\hbox{\Huge$\scriptstyle\cdot$}}}{\vbox{\hbox{\Large$\scriptstyle\cdot$}}}{\vbox{\hbox{\small$\scriptstyle\cdot$}}}},\widehat{S}^{-1}g)}$
Figure 2. Weighted version of the commutative diagram

The spaces $L^{p,q}(\mathbb{R}^{2d})$ and $L^{p,q}_{m}(\mathbb{R}^{2d})$ are
obviously isomorphic via

(4.1) $\Phi_{m}:L^{p,q}_{m}(\mathbb{R}^{2d})\to L^{p,q}(\mathbb{R}^{2d}),\quad
f\mapsto f\cdot m.$

In contrast to the pair
$(L^{p,q}(\mathbb{R}^{2d}),L^{p,q}_{m}(\mathbb{R}^{2d}))$, the relationship
$(\mathrm{M}^{p,q}(\mathbb{R}^{d}),\mathrm{M}^{p,q}_{m}(\mathbb{R}^{d}))$ is
not that obvious, requiring the formulation and proof of fairly intricate
lifting theorems (cf. [8, 16, 17] for isomorphisms between weighted modulation
spaces). Abstract coorbit theory does imply that the modulation spaces
$\mathrm{M}^{p,q}_{m_{1}}(\mathbb{R}^{d})$,
$\mathrm{M}^{p,q}_{m_{2}}(\mathbb{R}^{d})$ are isomorphic for two moderate
weights $m_{1},\,m_{2}$, but this observation provides little help when it
comes to the particular test we have on the block structure of the projection
of $\widehat{S}$.

Before we turn to these rather subtle questions, we first focus on the purely
measure-theoretic case, i.e., question whether the dilation operator acting on
the weighted space

(4.2) $(\mathcal{D}_{S})_{m}:L^{p,q}_{m}(\mathbb{R}^{2d})\to
L^{p,q}_{m}(\mathbb{R}^{2d}),\qquad f\mapsto f\circ S^{-1}$

is bounded. Using the isomorphism $\Phi_{m}$ from above, we compute

(4.3) $\Phi_{m}\ (\mathcal{D}_{S})_{m}\
\Phi_{m}^{-1}f=m\cdot\left(\left(\tfrac{f}{m}\right)\circ
S^{-1}\right)=m\cdot\ \frac{f\circ S^{-1}}{m\circ S^{-1}}=\frac{m}{m\circ
S^{-1}}\cdot\mathcal{D}_{S}f,$

where, as in the previous section, $\mathcal{D}_{S}$ denotes the dilation
operator acting on $L^{p,q}(\mathbb{R}^{2d})$.

######  Theorem 4.1.

thm:multiplication_operator_M_m Let $m$ be a moderate weight function. Let
$S\in\mathrm{Sp}(2d,\mathbb{R})$ be given. The following statements hold:

  1. (a)

If $\mathcal{D}_{S}:L^{p,q}(\mathbb{R}^{2d})\to L^{p,q}(\mathbb{R}^{2d})$ is
bounded, then $(\mathcal{D}_{S})_{m}$ is bounded if and only if

(4.4) $R_{m}\coloneqq\operatorname*{ess\,sup}\limits_{z\in\mathbb{R}^{2d}}\
{\frac{m(z)}{m(S^{-1}z)}}<\infty.$

  2. (b)

If $\mathcal{D}_{S}:L^{p,q}(\mathbb{R}^{2d})\to L^{p,q}(\mathbb{R}^{2d})$ is
unbounded and

(4.5) $T_{m}\coloneqq\operatorname*{ess\,inf}\limits_{z\in\mathbb{R}^{2d}}\
\frac{m(z)}{m(S^{-1}z)}>0,$

then $(\mathcal{D}_{S})_{m}$ is unbounded.

###### Proof.

  1. (a)

If $\mathcal{D}_{S}$ is bounded, it is in fact an automorphism, by Theorem
LABEL:thm:classification. Then Equation (4.3) implies that
$(\mathcal{D}_{S})_{m}$ is bounded if and only if the multiplication operator

(4.6) $M_{m}:L^{p,q}\to L^{p,q},\qquad f\mapsto\frac{m}{m\circ
S^{-1}}\mathop{\mathchoice{\vbox{\hbox{\Huge$\cdot$}}}{\vbox{\hbox{\Huge$\cdot$}}}{\vbox{\hbox{\Large$\cdot$}}}{\vbox{\hbox{\small$\cdot$}}}}f$

is bounded. A straightforward, somewhat tedious computation shows that $M_{m}$
is bounded if and only if $R_{m}$ is finite.

  2. (b)

According to prop:closed_operator, there is a $f\in L^{p,q}(\mathbb{R}^{2d})$
with $\mathcal{D}_{S}f\notin L^{p,q}(\mathbb{R}^{2d})$. It holds

(4.7) $\left\lVert M_{m}\ \mathcal{D}_{S}f\right\rVert_{p,q}\geq
T_{m}\left\lVert\mathcal{D}_{S}f\right\rVert_{p,q}=\infty.$

∎

######  Remark.

If $\mathcal{D}_{S}$ is unbounded, but $T_{m}=0$, then it is in general
difficult to make a reasonable claim. One can find bounds $\eta_{r}>0$, such
that

$\frac{1}{\eta_{r}}<\frac{m}{m\circ S^{-1}}<\eta_{r}$

on $[-r,r]^{2n}$ (cf. [15, Lemma 11.1.1.]). The problem lies on the behaviour
of $\eta_{r}$ as $r$ tends to $\infty$. If it is particularly slow, then we
could expect $M_{m}\,\mathcal{D}_{S}$ to be unbounded. It can, though, also
happen that $\frac{m}{m\circ S}$ decays so fast, that it compensates for the
growth of $\mathcal{D}_{S}f$ and

(4.8) $M_{m}\,\mathcal{D}_{S}:L^{p,q}\to L^{p,q}$

is well-defined. According to prop:closed_operator, this would suffice for the
boundedness of $(\mathcal{D}_{S})_{m}$.

Observe that the case $m\asymp m\circ S^{-1}$ is equivalent to $0<T_{m}\leq
R_{m}<\infty$. In this case, the positive results from
thm:multiplication_operator_M_m yield the following corollary.

######  Corollary 4.2.

cor:equiv_weight Let $m$ be a moderate weight and
$S\in\mathrm{Sp}(2d,\mathbb{R})$ a symplectic matrix. Assuming $m\asymp m\circ
S^{-1}$, the operator

(4.9) $\mathcal{D}_{S}:L^{p,q}\to L^{p,q}$

is bounded if and only if the operator

(4.10) $(\mathcal{D}_{S})_{m}:L^{p,q}_{m}\to L^{p,q}_{m}$

is bounded.

In light of this, we prove an analogous version for weighted modulation
spaces.

###  4.2. The Lifting on $\mathrm{M}^{p,q}_{m}(\mathbb{R}^{d})$

To establish the connection between the metaplectic action on unweighted and
on weighted modulation spaces, we make use of _Toeplitz operators_.

######  Definition 4.3.

Let $a\in\mathcal{S}(\mathbb{R}^{2d})$ be a symbol and
$g\in\mathcal{S}(\mathbb{R}^{2d})$ a fixed window. Then the Toeplitz operator
$\mathrm{Tp}_{g}(a)$ is defined by the formula

(4.11) $\left\langle\mathrm{Tp}_{g}(a)f_{1},f_{2}\right\rangle_{\tiny
L^{2}(\mathbb{R}^{d})}=\left\langle
a\mathcal{V}_{g}f_{1},\mathcal{V}_{g}f_{2}\right\rangle_{\tiny
L^{2}(\mathbb{R}^{2d})}=\left\langle
a\,\mathrm{A}(f_{1},g),\mathrm{A}(f_{2},g)\right\rangle_{\tiny
L^{2}(\mathbb{R}^{2d})}$

for all $f_{1},f_{2}\in L^{2}(\mathbb{R}^{d})$. It is a well-defined operator
which extends uniquely to a continuous operator from
$\mathcal{S}^{\prime}(\mathbb{R}^{d})$ to $\mathcal{S}(\mathbb{R}^{d})$. The
class of admissible symbols can be significantly extended (cf. [16,
Proposition 1.5.]). In particular, we can choose the symbol to be our weight
function.

######  Theorem 4.4 (Gröchenig, Toft [16]).

Assume that $m$ is an even, $v$-moderate weight function, for the weight
$v(x)=(1+\|x\|)^{N}$, and $g\in\mathcal{S}(\mathbb{R}^{d})$. Then the Toeplitz
operator $\mathrm{Tp}_{g}(m)$ is an isomorphism from
$\mathrm{M}^{p,q}_{m_{0}}(\mathbb{R}^{d})$ onto
$\mathrm{M}^{p,q}_{m_{0}/m}(\mathbb{R}^{d})$ for every $v$-moderate even
weight $m_{0}$ and every $p,q\in[1,\infty]$.

We will apply this theorem with $m_{0}=m$. Before that, we rewrite the
defining property of $\mathrm{Tp}_{g}(a)f$ in terms of its ambiguity function.
We recall that the standard symplectic form
$[\gamma,\lambda]=\gamma\cdot\mathcal{J}\lambda$ on $\mathbb{R}^{2d}$ and the
symmetric time-frequency shifts satisfy

(4.12) $\rho(\lambda)\rho(\gamma)=\rho(\lambda+\gamma)e^{\pi
i[\gamma,\lambda]}.$

From the definition of the ambiguity function, one easily sees that
$\overline{\mathrm{A}(g,g)(\lambda)}=\mathrm{A}(g,g)(-\lambda)$ holds for all
$\lambda\in\mathbb{R}^{2d}$. Let now $g\in\mathcal{S}(\mathbb{R}^{d}),\,g\neq
0$ be an arbitrary window and $a:\mathbb{R}^{2d}\to\mathbb{C}$ a symbol. Then
it holds

(4.13)
$\begin{split}A(\mathrm{Tp}_{g}(a)f,g)(\lambda)&=\left\langle\mathrm{Tp}_{g}(a)f,\rho(\lambda)g\right\rangle\\\
&=\left\langle a\mathrm{A}(f,g),\mathrm{A}(\rho(\lambda)g,g)\right\rangle\\\
&=\int_{\mathbb{R}^{2d}}a(\gamma)\mathrm{A}(f,g)(\gamma)\overline{\left\langle\rho(\lambda)g,\rho(\gamma)g\right\rangle}\,d\gamma\\\
&=\int_{\mathbb{R}^{2d}}a(\gamma)\mathrm{A}(f,g)(\gamma)\overline{\left\langle
g,\rho(-\lambda)\rho(\gamma)g\right\rangle}\,d\gamma\\\
&=\int_{\mathbb{R}^{2d}}a(\gamma)\mathrm{A}(f,g)(\gamma)\overline{\left\langle
g,\rho(-\lambda+\gamma)g\right\rangle e^{-\pi i[\gamma,\lambda]}}\,d\gamma\\\
&=\int_{\mathbb{R}^{2d}}a(\gamma)\mathrm{A}(f,g)(\gamma)e^{\pi
i[\gamma,\lambda]}\mathrm{A}(g,g)(\lambda-\gamma)e^{\pi
i[\gamma,\lambda]}\,d\gamma.\end{split}$

This shows that the Toeplitz operator can be understood through the _twisted
convolution_. The twisted convolution $F\natural G$ of two measurable
functions $F,\,G:\mathbb{R}^{2d}\to\mathbb{C}$ is given by

(4.14) $F\natural
G(\lambda)\coloneqq\int_{\mathbb{R}^{2d}}F(\gamma)G(\lambda-\gamma)e^{\pi
i[\gamma,\lambda]}\ d\gamma,$

assuming the integral is well-defined. In terms of the Toeplitz operator
$\mathrm{Tp}_{g}(a)$, we can now express its action as

(4.15)
$\mathrm{A}(\mathrm{Tp}_{g}(a)f,g)=\left(a\,\mathrm{A}(f,g)\right)\natural\mathrm{A}(g,g).$

######  Corollary 4.5.

Let $m$ be a moderate weight, $p,q\in[1,\infty]$. If
$\widehat{S}:\mathrm{M}^{p,q}(\mathbb{R}^{d})\to\mathrm{M}^{p,q}(\mathbb{R}^{d})$
is bounded and

$C_{m}\coloneqq\operatorname*{ess\,sup}\limits_{z\in\mathbb{R}^{2d}}{\frac{m(z)}{m(S^{-1}z)}}<\infty,$

then

$\widehat{S}:\mathrm{M}^{p,q}_{m}(\mathbb{R}^{d})\to\mathrm{M}^{p,q}_{m}(\mathbb{R}^{d})$

is well-defined and bounded.

###### Proof.

Theorems LABEL:thm:classification and LABEL:thm:multiplication_operator_M_m
imply that $(\mathcal{D}_{S})_{m}:L^{p,q}_{m}\to L^{p,q}_{m}$ is bounded. The
commutative diagram completes the proof. ∎

We now focus on the opposite implication.

######  Theorem 4.6.

thm:main_weighted Let $m$ be a moderate weight, $p,q\in[1,\infty]$. If
$\widehat{S}\in\mathrm{Mp}(2d,\mathbb{R})$ with projection $\pi^{\tiny
Mp}(\widehat{S})=S$ satisfies $m\asymp m\circ S^{-1}$, then $\widehat{S}$ is a
bounded operator from $\mathrm{M}^{p,q}(\mathbb{R}^{d})$ to
$\mathrm{M}^{p,q}(\mathbb{R}^{d})$ if and only if it is a bounded operator
from $\mathrm{M}^{p,q}_{m}(\mathbb{R}^{d})$ to
$\mathrm{M}^{p,q}_{m}(\mathbb{R}^{d})$.

###### Proof.

The first implication follows from the last corollary. Let now $\widehat{S}$
be a bounded operator from $\mathrm{M}^{p,q}_{m}(\mathbb{R}^{d})$ to
$\mathrm{M}^{p,q}_{m}(\mathbb{R}^{d})$. We fix as a window the normalized
Gaussian $g\in\mathcal{S}(\mathbb{R}^{d})$ and recall the symplectic
covariance of the ambiguity function (2.21). For all
$f\in\mathrm{M}^{p,q}(\mathbb{R}^{d})$ holds

(4.16)
$\begin{split}\left\lVert\widehat{S}f\right\rVert_{\mathrm{M}^{p,q}(\mathbb{R}^{d})}&=\left\lVert\mathrm{A}(\widehat{S}f,g)\right\rVert_{L^{p,q}}\overset{\scriptscriptstyle\tiny\text{Lift}}{\underset{\tiny\eqref{eq:TpTwisted}}{\asymp}}\left\lVert\left(\tfrac{1}{m}\mathrm{A}(\widehat{S}f,g)\right)\
\natural\ \mathrm{A}(g,g)\right\rVert_{L^{p,q}_{m}}\\\\[4.30554pt]
\overset{\eqref{eq:symplectic_covariance}}{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}=}}&=\left\lVert\left(\tfrac{1}{m}\mathcal{D}_{S}\mathrm{A}(f,\widehat{S}^{-1}g)\right)\
\natural\ \mathrm{A}(g,g)\right\rVert_{L^{p,q}_{m}}\\\\[4.30554pt]
&=\left\lVert\left(\tfrac{m\circ S^{-1}}{m\circ S^{-1}}\tfrac{1}{\left(m\circ
S\circ\right)S^{-1}}\mathcal{D}_{S}\ \mathrm{A}(f,\widehat{S}^{-1}g)\right)\
\natural\ \mathrm{A}(g,g)\right\rVert_{L^{p,q}_{m}}\\\\[4.30554pt]
&=\left\lVert\left((\mathcal{D}_{S})_{m\widetilde{m}^{-1}}\
(\widetilde{m}m^{-1}\mathrm{A}(f,\widehat{S}^{-1}g))\right)\ \natural\
\mathrm{A}(g,g)\right\rVert_{L^{p,q}_{m}},\\\
&=\left\lVert(\mathcal{D}_{S})_{m\widetilde{m}^{-1}}(\mathcal{D}_{S})_{m\widetilde{m}^{-1}}^{-1}\;\left(\left((\mathcal{D}_{S})_{m\widetilde{m}^{-1}}\
(\widetilde{m}m^{-1}\mathrm{A}(f,\widehat{S}^{-1}g))\right)\ \natural\
\mathrm{A}(g,g)\right)\right\rVert_{L^{p,q}_{m}},\end{split}$

where $\widetilde{m}=\tfrac{m}{\mathcal{D}_{S}^{-1}m}$. Recall that the
symplectic matrices are exactly those which preserve the standard symplectic
form. Therefore, the twisted convolution of two functions satisfies

(4.17) $\begin{split}\left(F\natural
G\right)(S^{-1}\lambda)&=\int_{\mathbb{R}^{2d}}F(\gamma)G(S^{-1}\lambda-\gamma)e^{\pi
i\sigma(\gamma,S^{-1}\lambda)}\ d\gamma\\\\[4.30554pt]
&=\int_{\mathbb{R}^{2d}}F(S^{-1}S\gamma)G(S^{-1}(\lambda-S\gamma))e^{\pi
i\sigma(S\gamma,SS^{-1}\lambda)}\ d\gamma\\\\[4.30554pt]
=&\int_{\mathbb{R}^{2d}}F(S^{-1}\gamma)G(S^{-1}(\lambda-\gamma))e^{\pi
i\sigma(\gamma,\lambda)}\ d\gamma=(F\circ S^{-1})\,\natural\,(G\circ
S^{-1})(\lambda).\end{split}$

This, along with the symplectic covariance of the ambiguity function (2.21),
leads to

(4.18)
$\begin{split}\left\lVert\widehat{S}f\right\rVert_{\mathrm{M}^{p,q}(\mathbb{R}^{d})}\asymp&\left\lVert(\mathcal{D}_{S})_{m\widetilde{m}^{-1}}\;\left(\left(\widetilde{m}m^{-1}\cdot\mathrm{A}(f,\widehat{S}^{-1}g)\right)\
\natural\
\mathrm{A}(\widehat{S}^{-1}g,\widehat{S}^{-1}g)\right)\right\rVert_{L^{p,q}_{m}}.\end{split}$

We recall that $\widehat{S}^{-1}g$ is a Schwartz function. By the lifting
Theorem 4.4 and Equation (4.15), there exists a tempered distribution
$\widetilde{f}\in\mathrm{M}^{p,q}_{\widetilde{m}^{-1}m}(\mathbb{R}^{d})$ with

(4.19) $\left(\widetilde{m}m^{-1}\cdot\mathrm{A}(f,\widehat{S}^{-1}g)\right)\
\natural\
\mathrm{A}(\widehat{S}^{-1}g,\widehat{S}^{-1}g)=\mathrm{A}(\widetilde{f},\widehat{S}^{-1}g).$

Recall that Theorem 4.4 also provides the norm equivalence
$\|\tilde{f}\|_{M^{p,q}_{\widetilde{m}^{-1}m}}\asymp\|f\|_{M^{p,q}}$, where
the implied constant is independent of $f\in\mathrm{M}^{p,q}(\mathbb{R}^{d})$.
By assumption, $\widetilde{m}\asymp 1$, which implies
$\mathrm{M}^{p,q}_{\widetilde{m}^{-1}m}(\mathbb{R}^{d})=\mathrm{M}^{p,q}_{m}(\mathbb{R}^{d})$
and

(4.20)
$\left\lVert\widetilde{f}\right\rVert_{\mathrm{M}^{p,q}_{m}(\mathbb{R}^{d})}\asymp\left\lVert\widetilde{f}\right\rVert_{\mathrm{M}^{p,q}_{\widetilde{m}^{-1}m}(\mathbb{R}^{d})}\asymp\left\lVert
f\right\rVert_{\mathrm{M}^{p,q}(\mathbb{R}^{d})}.$

Putting it all together, we obtain

(4.21)
$\begin{split}\left\lVert\widehat{S}f\right\rVert_{\mathrm{M}^{p,q}(\mathbb{R}^{d})}\asymp&\left\lVert(\mathcal{D}_{S})_{m}\mathrm{A}(\widetilde{f},\widehat{S}^{-1}g)\right\rVert_{L^{p,q}_{m}}=\left\lVert\mathrm{A}(\widehat{S}\widetilde{f},g)\right\rVert_{L^{p,q}_{m}}\\\\[4.30554pt]
\asymp&\left\lVert\widehat{S}\widetilde{f}\right\rVert_{\mathrm{M}^{p,q}_{m}(\mathbb{R}^{d})}\lesssim\left\lVert\widetilde{f}\right\rVert_{\mathrm{M}^{p,q}_{m}(\mathbb{R}^{d})}\asymp\left\lVert
f\right\rVert_{\mathrm{M}^{p,q}(\mathbb{R}^{d})}.\end{split}$

This proves the claim. ∎

###  4.3. Examples

We consider two of the standard weights which are frequently used in time-
frequency analysis and determine the cases where our criteria are applicable.
We refer to [12] for a general overview of typical weights.

######  Example A.

We begin with

(4.22) $m(z)=m_{s,t}(z)=(1+\left\lVert z\right\rVert)^{s}(\ln(e+\left\lVert
z\right\rVert))^{t}.$

This class contains in particular the radial polynomial weights. Here it is
well-known that $\|z\|\asymp\|Sz\|$ holds for any invertible matrix
$S\in\mathbb{R}^{2d\times 2d}$. This immediately shows the equivalence
$m_{s,0}\asymp m_{s,0}\circ S$.

For the logarithmic part, we spell out the norm equivalence as

$A\|z\|\leq\|Sz\|\leq B\|z\|~{},$

with $0<A\leq B$, and obtain

(4.23) $\begin{split}\frac{\ln(e+\left\lVert
z\right\rVert)}{\ln(e+B\left\lVert z\right\rVert)}\leq\frac{\ln(e+\left\lVert
z\right\rVert)}{\ln(e+\left\lVert Sz\right\rVert)}\leq\frac{\ln(e+\left\lVert
z\right\rVert)}{\ln(e+A\left\lVert z\right\rVert)},\end{split}$

We next establish bounds for the ends of this chain of inequalities as
$\|z\|\to 0$ and $\|z\|\to\infty$. Using l’Hospital’s rule allows computing
the limits

(4.24) $\lim\limits_{z\to 0}\frac{\ln(e+\left\lVert
z\right\rVert)}{\ln(e+A\left\lVert z\right\rVert)}=\lim\limits_{z\to
0}\frac{\ln(e+\left\lVert z\right\rVert)}{\ln(e+B\left\lVert
z\right\rVert)}=1~{},$

as well as

(4.25) $\lim\limits_{z\to\infty}\frac{\ln(e+\left\lVert
z\right\rVert)}{\ln(e+A\left\lVert
z\right\rVert)}=\lim\limits_{z\to\infty}\frac{\ln(e+\left\lVert
z\right\rVert)}{\ln(e+B\left\lVert z\right\rVert)}=1~{}.$

As a consequence there exists $\varepsilon>0$ such that

$\frac{1}{2}<\frac{\ln(e+\left\lVert z\right\rVert)}{\ln(e+\left\lVert
Sz\right\rVert)}<2$

holds for all $z\in\mathbb{R}^{2d}$ satisfying either $\left\lVert
z\right\rVert<\epsilon$ or $\left\lVert z\right\rVert>\epsilon^{-1}$. However,
the remaining $z$ constitute a compact set, and
$z\mapsto\frac{\ln(e+\left\lVert z\right\rVert)}{\ln(e+\left\lVert
Sz\right\rVert)}$ is a nonvanishing continuous function, hence there exist
nontrivial lower and upper bounds valid on this set. This shows $m_{0,t}\asymp
m_{0,t}\circ S$, for all $t$.

Combining our results so far, we obtain for fixed $s,t$ that

(4.26) $\frac{m_{s,t}}{m_{s,t}\circ S}\asymp 1~{}.$

Hence cor:equiv_weight together with thm:classification provides the
conclusion that the operator
$\widehat{S}:\mathrm{M}^{p,q}_{m_{s,t}}(\mathbb{R}^{d})\to\mathrm{M}^{p,q}_{m_{s,t}}(\mathbb{R}^{d})$
is bounded iff $p=q$, or $p\not=q$ and $S$ is upper block triangular. Since
the latter criterion applies to $S$ iff it applies to its inverse, we further
obtain that boundedness of $\widehat{S}$ on
$\mathrm{M}^{p,q}_{m_{s,t}}(\mathbb{R}^{d})$ already implies that it is an
automorphism.

We now turn to the second type of polynomial weight functions, where the
condition $m\asymp m\circ S$ is actually restrictive.

######  Example B.

According to [15, Ex. 11.1.1.], it holds

(4.27) $(1+\left\lVert z\right\rVert)^{s}\asymp(1+\left\lVert
x\right\rVert+\left\lVert\omega\right\rVert)^{s},\qquad
x,\omega\in\mathbb{R}^{d},\,z=(x,\omega).$

This weight is symmetric in $(x,\omega)$. However, sometimes, we are only
interested in the growth in one direction. This motivates a different class of
weight functions, given by

(4.28) $\begin{split}p_{s}(x,\omega)&=p_{s}(x)=(1+\left\lVert
x\right\rVert)^{s},\qquad x,\omega\in\mathbb{R}^{d},\\\
q_{t}(x,\omega)&=q_{t}(\omega)=(1+\left\lVert\omega\right\rVert)^{t},\qquad
x,\omega\in\mathbb{R}^{d},\end{split}$

with $s,t\in\mathbb{R}$. Since we are only interested in nontrivial weights,
we assume $s\not=0\not=t$ throughout. We use the block-matrix notation

(4.29) $S=\begin{pmatrix}A&B\\\ C&D\end{pmatrix}\qquad\text{and}\qquad
z=\begin{pmatrix}x\\\ \omega\end{pmatrix},$

and get

(4.30) $\frac{(p_{s}\circ S)(z)}{p_{s}(z)}=\left(\frac{1+\left\lVert
Ax+B\omega\right\rVert}{1+\left\lVert x\right\rVert}\right)^{s}$

We first show that $p_{s}\circ S\asymp p_{s}$ holds iff $S$ is lower block
triangular, i.e., iff $B=0$.

To see the ”only-if” part, assume $B\neq 0$ and $s>0$. There is
$\omega\in\mathbb{R}^{d}$ with $\left\lVert B\omega\right\rVert>\left\lVert
Ae_{1}\right\rVert$. Therefore,

(4.31) $\left(\frac{1+\left\lVert Ae_{1}+B\,n\omega\right\rVert}{1+\left\lVert
e_{1}\right\rVert}\right)^{s}\geq\left(\frac{1+n\left\lVert
B\omega\right\rVert-\left\lVert
Ae_{1}\right\rVert}{2}\right)^{s}\to\infty\quad(n\to\infty).$

In the case $s<0$, we observe that if $S$ is not a lower block triangular
matrix, the same applies to $S^{-1}$. Using $z^{\prime}=Sz$, we can use the
previous calculation to establish the unboundedness of
$\frac{p_{s}}{p_{s}\circ S}$. Hence we have $p_{s}\circ S\not\asymp p_{s}$ for
al $s\not=0$.

Conversely, if $B=0$, $A$ is invertible, and we have the well-known fact that

(4.32) $\frac{p_{s}\circ S}{p_{s}}(x,\omega)=\left(\frac{1+\left\lVert
Ax\right\rVert}{1+\left\lVert x\right\rVert}\right)^{s}\asymp 1~{}.$

This implies that in the case where $S$ is not lower block triangular, neither
thm:multiplication_operator_M_m nor thm:main_weighted is applicable, leaving
the question of boundedness of $\widehat{S}$ on
$\mathrm{M}^{p,q}_{p_{s}}(\mathbb{R}^{d})$ wide open.

In the case where $S$ is lower block triangular, the combination of
cor:equiv_weight and thm:classification yields that $\widehat{S}$ is bounded
iff $S$ is upper block triangular, or equivalently, iff $S$ is block-diagonal.

For the weight $q_{t}$ with $t\not=0$, the analogous reasoning as for $p_{s}$
yields that $q_{t}\circ S\asymp q_{t}$ holds iff $S$ is upper block
triangular. Again, this leaves the question of boundedness of $\widehat{S}$
open for matrices that are not upper block triangular. However, for matrices
$S$ that are upper block triangular, cor:equiv_weight together with
thm:classification already implies the boundedness of $\widehat{S}$ on
$\mathrm{M}^{p,q}_{q_{t}}(\mathbb{R}^{d})$, without further restrictions on
$S$.

## Concluding remarks

Our results concerning the mapping behavior of metaplectic operators
$\widehat{S}$ on unweighted modulation spaces give a complete picture. To our
knowledge, even the mapping properties of the associated dilation operators
$\mathcal{D}_{S}:L^{p,q}(\mathbb{R}^{2d})\to L^{p,q}(\mathbb{R}^{2d})$ were
not fully clarified prior to our paper, and the parts of Theorem
LABEL:thm:main_unweighted dealing with this problem seem to be new.

The weighted case poses several novel challenges. The first one concerns the
choice of weights: Our paper concentrates on even polynomial weights, in order
to apply results from [16], and we are not aware of extensions of these
results to other weight classes. Note that the arguments of Gröchenig and Toft
rely on some fairly subtle properties of algebras of symbol classes, and to
us, it seems open whether generalizations of the desired type are available.

The second unresolved question concerns the mapping properties of metaplectic
operators
$\widehat{S}:\mathrm{M}^{p,q}_{m}(\mathbb{R}^{d})\to\mathrm{M}^{p,q}_{m}(\mathbb{R}^{d})$
in the case where $\widehat{S}$ is unbounded on the unweighted modulation
space $\mathrm{M}^{p,q}(\mathbb{R}^{d})$, and the weights $m$ and $m\circ
S^{-1}$ are not equivalent. We presented examples that this case can occur for
fairly natural choices of weights. Understanding the associated measure-
theoretic question, concerning the boundedness properties of the dilation
operator $\mathcal{D}_{S}:L^{p,q}_{m}(\mathbb{R}^{2d})\to
L^{p,q}_{m}(\mathbb{R}^{2d})$ clearly constitutes an important step towards
answering the question in full.

##  Appendix A Calculation for Equation (3.16)

To prove the unboundedness of $\mathcal{D}_{S}$ when $S$ is not an upper block
triangular matrix, we need the following technical lemma.

######  Lemma A.1.

Let $A=A^{T}\in\mathbb{R}^{n\times n}$ and $\beta\in\mathbb{R}^{d}$ be given.
It holds

(A.1) $\mathcal{I}\coloneqq\int_{\mathbb{R}^{d}}\exp(-\pi x\cdot
Ax+2\pi\beta\cdot x)\ dx=\left\lvert\det A\right\rvert^{-\frac{1}{2}}\
\exp(\pi\beta\cdot A^{-1}\beta),$

where the integral converges if and only if $A$ is strictly positive definite.

###### Proof.

Let $A=U^{T}\Lambda U$ with $U$ orthogonal und $\Lambda$ diagonal. Recall that
the Lebesgue measure is invariant under the action of the Euclidean group. The
integrand is non-negative, so $\mathcal{I}\in[0,\infty]$ is well-defined.

(A.2) $\mathcal{I}=\int_{\mathbb{R}^{d}}\exp(-\pi(Ux)\cdot\Lambda
Ux+2\pi\beta\cdot U^{T}Ux)\ dx=\int_{\mathbb{R}^{d}}\exp(-\pi x\cdot\Lambda
x+2\pi(U\beta)\cdot x)\ dx.$

The integral

(A.3) $\int_{\mathbb{R}}e^{-at^{2}+bt}dt<\infty$

is finite if and only if $a>0$. Hence $\mathcal{I}<\infty$ if and only if
$\Lambda$ is strictly positive definite.

(A.4)
$\begin{split}\mathcal{I}&=\int_{\mathbb{R}^{d}}\exp\Big{(}-\pi(\Lambda^{\frac{1}{2}}x)\cdot(\Lambda^{\frac{1}{2}}x)+2\pi(\Lambda^{-\frac{1}{2}}U\beta)\cdot\
\Lambda^{\frac{1}{2}}x\\\
&\qquad\qquad-\pi(\Lambda^{-\frac{1}{2}}U\beta)\cdot\
(\Lambda^{-\frac{1}{2}}U\beta)+\pi(\Lambda^{-\frac{1}{2}}U\beta)\cdot\
(\Lambda^{-\frac{1}{2}}U\beta)\Big{)}\ dx\\\ {=}&\left\lvert\det\
\Lambda^{\frac{1}{2}}\right\rvert^{-1}\int_{\mathbb{R}^{d}}\exp\Big{(}-\pi
x\cdot x+\pi(\Lambda^{-\frac{1}{2}}U\beta)\cdot\
(\Lambda^{-\frac{1}{2}}U\beta)\Big{)}\ dx\\\ &=\left\lvert\det\ U^{T}\Lambda
U\right\rvert^{-\frac{1}{2}}\ \exp(\pi\beta\cdot
U^{T}\Lambda^{-\frac{1}{2}}\Lambda^{-\frac{1}{2}}U\beta)=\left\lvert\det\
A\right\rvert^{-\frac{1}{2}}\ \exp(\pi\beta\cdot A^{-1}\beta).\end{split}$

∎

We now compute the ambiguity function of the Gaussian.

(A.5) $\begin{split}&\mathrm{A}\left(\
g\circ(\varepsilon^{2}-1)^{\frac{1}{2}}I,g\ \right)(x,\omega)\\\ &=e^{\pi
ix\cdot\omega}\int_{\mathbb{R}^{d}}\exp\left(-\pi
t\cdot(E^{2}-I)t-\pi(t-x)\cdot(t-x)-2\pi i\omega\cdot t\right)\ dt\\\ &=e^{\pi
ix\cdot\omega}\int_{\mathbb{R}^{d}}\exp\left(-\pi t\cdot E^{2}t+2\pi x\cdot
t-\pi x\cdot x-2\pi i\omega\cdot t\right)\ dt\\\ =&\;e^{\pi
ix\cdot\omega}\left\lvert\det\
E\right\rvert^{-1}\int_{\mathbb{R}^{d}}\exp\Big{(}-\pi t\cdot
t+2\pi(E^{-1}x)\cdot t-\pi x\cdot E^{-2}x\\\ &\qquad\qquad-\pi
x\cdot(I-E^{-2})x-2\pi i(E^{-1}\omega)\cdot(t-E^{-1}x)-2\pi
i(E^{-1}\omega)\cdot E^{-1}x\Big{)}\ dt\\\\[8.61108pt]
\overset{\scriptscriptstyle T_{E^{-1}x}}{=}&\;\left\lvert\det\
E\right\rvert^{-1}e^{-\pi x\cdot(I-E^{-2})x-2\pi i(E^{-1}\omega)\cdot
E^{-1}x+\pi ix\cdot\omega}\cdot\int_{\mathbb{R}^{d}}\exp\left(-\pi t\cdot
t-2\pi i(E^{-1}\omega)\cdot t\right)\ dt\\\
\overset{\scriptscriptstyle\mathcal{F}g=g}{=}&\;\left\lvert\det\
E\right\rvert^{-1}e^{-\pi x\cdot(I-E^{-2})x-2\pi i(E^{-1}\omega)\cdot
E^{-1}x+\pi ix\cdot\omega-\pi\omega\cdot E^{-2}\omega}\\\
\overset{\scriptscriptstyle\Delta^{2}+E^{-2}=I}{=}&\;\left\lvert\det\
E\right\rvert^{-1}e^{\pi ix\cdot\omega-2\pi i\omega\cdot E^{-2}x}e^{-\pi
x\cdot\Delta^{2}x}e^{-\pi\omega\cdot E^{-2}\omega}.\end{split}$

##  Appendix B Calculation for Equation (3.20)

We are concerned with the $L^{p,q}$ norm of

(B.1)
$\mathcal{D}_{S}f(x,\omega)=\exp\left(-\pi(Ax+B\omega)\cdot\Delta^{2}(Ax+B\omega)-\pi(Cx+D\omega)\cdot
E^{-2}(Cx+D\omega)\right).$

To this end, we group the scalar products:

(B.2) $\displaystyle\ \quad x\cdot A^{T}\Delta^{2}Ax+2\omega\cdot
B^{T}\Delta^{2}Ax+\omega\cdot B^{T}\Delta^{2}B\omega$ (B.3)
$\displaystyle\qquad\qquad+x\cdot C^{T}E^{-2}C^{T}x+2\omega\cdot
D^{T}E^{-2}Cx+\omega\cdot D^{T}E^{-2}D\omega$ (B.4)
$\displaystyle=x\cdot(A^{T}\Delta^{2}A+C^{T}E^{-2}C)x+2\omega\cdot(B^{T}\Delta^{2}A+D^{T}E^{-2}C)x+\omega\cdot(B^{T}\Delta^{2}B+D^{T}E^{-2}D)\omega.$

For a better overview, we define

(B.5) $\begin{split}\Sigma&\coloneqq A^{T}\Delta^{2}A+C^{T}E^{-2}C,\\\
\beta&\coloneqq B^{T}\Delta^{2}A+D^{T}E^{-2}C\\\ &\text{and}\\\
\Omega&\coloneqq
B^{T}\Delta^{2}B+D^{T}E^{-2}D-\beta\Sigma^{-1}\beta^{T}.\end{split}$

With the new notation, it holds

(B.6)
$\left\lVert\mathcal{D}_{S}f\right\rVert_{{p,q}}=\left\lVert\omega\mapsto
e^{-\pi\omega\cdot(\Omega+\beta\Sigma^{-1}\beta^{T})\omega}\cdot\left\lVert
x\mapsto e^{-\pi x\cdot\Sigma x-2\pi\omega\cdot\beta
x}\right\rVert_{p}\right\rVert_{q}.$

By the definition of $\Sigma$, its spectrum is a subset of $[0,\infty)$. In
the cases that we will observe, the spectrum of $\Sigma$ and the spectrum of
$\Omega$ will lie in $(0,\infty)$. This is no accident, since
$\mathrm{A}(\varphi,g)\in\mathcal{S}(\mathbb{R}^{d})$ for all
$\varphi\in\mathcal{S}(\mathbb{R}^{d})\subset\mathrm{M}^{p,q}(\mathbb{R}^{d})$.
If $p<\infty$, the inner integral equals

(B.7) $\begin{split}n(\omega)&=\left\lVert x\mapsto e^{-p\pi x\cdot\Sigma
x-2p\pi\omega\cdot\beta
x}\right\rVert_{1}^{\frac{1}{p}}\overset{\scriptscriptstyle\ref{lemma:NormFormel}}{=}\left\lvert\det\
p\Sigma\right\rvert^{-\frac{1}{2p}}\
\left(e^{p\pi(\omega\cdot\beta)\Sigma^{-1}\beta^{T}\omega}\right)^{\frac{1}{p}}\\\
&{\asymp}\left\lvert\det\
\Sigma\right\rvert^{-\frac{1}{2p}}e^{\pi\omega\cdot\beta\Sigma^{-1}\beta^{T}\omega}.\end{split}$

If $p=\infty$, then the function

(B.8) $F(x,\omega)\coloneqq e^{-\pi x\cdot\Sigma x-2\pi\omega\cdot\beta
x}=e^{-\pi(x+\Sigma^{-1}\beta^{T}\omega)\cdot\Sigma(x+\Sigma^{-1}\beta^{T}\omega)+\pi\omega\cdot\beta\Sigma^{-1}\beta^{T}\,\omega}\leq
e^{\pi\omega\cdot\beta\Sigma^{-1}\beta^{T}\,\omega}~{},$

with equality if and only if $x=-\Sigma^{-1}\beta^{T}\,\omega$. Hence, for all
$p\in[1,\infty]$ holds

(B.9)
$\begin{split}\left\lVert\mathcal{D}_{S}f\right\rVert_{{p,q}}&\asymp\left\lvert\det\
\Sigma\right\rvert^{-\frac{1}{2p}}\left\lVert\omega\mapsto
e^{-\pi\omega\cdot(\Omega+\beta\Sigma^{-1}\beta^{T})\omega}\cdot
e^{\pi\omega\cdot\beta\Sigma^{-1}\beta^{T}\omega}\right\rVert_{q}\\\
&=\left\lvert\det\ \Sigma\right\rvert^{-\frac{1}{2p}}\left\lVert\omega\mapsto
e^{-\pi\omega\cdot\Omega\omega}\right\rVert_{q}.\end{split}$

For the case $q=\infty$ it immediately follows that the remaining norm is $1$.
For all other $q\in[1,\infty)$ holds

(B.10) $\left\lVert\mathcal{D}_{S}f\right\rVert_{{p,q}}\asymp\left\lvert\det\
\Sigma\right\rvert^{-\frac{1}{2p}}\ \left\lvert\det(q\
\Omega)\right\rvert^{-\frac{1}{2q}}\asymp\left\lvert\det\
\Sigma\right\rvert^{-\frac{1}{2p}}\left\lvert\det\
\Omega\right\rvert^{-\frac{1}{2q}}.$

All in all,

(B.11) $\left\lVert\mathcal{D}_{S}f\right\rVert_{{p,q}}\asymp\left\lvert\det\
\Sigma\right\rvert^{-\frac{1}{2p}}\left\lvert\det\
\Omega\right\rvert^{-\frac{1}{2q}}$

for all $p,q\in[1,\infty]$.

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