# An excursion to multiplications and convolutions on modulation spaces

Nenad Teofanov  Department of Mathematics and Informatics, University of Novi
Sad, Novi Sad, Serbia<EMAIL_ADDRESS>and  Joachim Toft
Department of Mathematics, Linnæus University, Växjö, Sweden
<EMAIL_ADDRESS>
###### Abstract.

We give a self-contained introduction to (quasi-)Banach modulation spaces of
ultradistributions, and review results on boundedness for multiplications and
convolutions for elements in such spaces. Furthermore, we use these results to
study the Gabor product. As an example, we show how it appears in a phase-
space formulation of the nonlinear cubic Schrödinger equation.

###### Key words and phrases:

time–frequency analysis, modulation spaces, convolutions, multiplications

###### 2010 Mathematics Subject Classification:

42B35, 44A15, 46A16, 16W80

##  0\. Introduction

Modulation spaces were introduced in Feichtinger’s seminal technical report
[17], and prove themselves as useful family of Banach spaces of tempered
distributions in time-frequency analysis, [6, 10, 29]. The main purpose of
this survey article is to enlighten some properties of modulation spaces in a
rather self-contained manner. In contrast to the most common situation, our
analysis includes both quasi-Banach and Banach modulation spaces within the
framework of ultradifferentiable functions and ultradistributions of
Gelfand–Shilov type. For that reason we collect necessary background material
in a rather detailed preliminary section.

Motivated by recent applications of modulation spaces in the context of
nonlinear harmonic analysis and its applications, cf. [3, 4, 6, 14, 23, 39,
40, 48, 55] we focus our attention to boundedness for multiplications and
convolutions for elements in such spaces. The basic results in that direction
go back to the original contribution [17], and were thereafter reconsidered by
many authors in different contexts. Let us give a brief, and unavoidably
incomplete account on the related results.

In Section 2 we formulate in Theorems 2.5 and 2.7 bilinear versions of more
general multiplication and convolution results in [55, Section 3]. The
contents of Theorems 2.5 and 2.7 in the unweighted case for modulation spaces
$M^{p,q}$ can be summarized as follows.

######  Proposition 0.1.

Let $p_{j},q_{j}\in(0,\infty]$, $j=0,1,2$,

$\theta_{1}=\max\left(1,\frac{1}{p_{0}},\frac{1}{q_{1}},\frac{1}{q_{2}}\right)\quad\text{and}\quad\theta_{2}=\max\left(1,\frac{1}{p_{1}},\frac{1}{p_{2}}\right).$

Then

$\displaystyle M^{p_{1},q_{1}}\cdot M^{p_{2},q_{2}}$ $\displaystyle\subseteq
M^{p_{0},q_{0}},$ $\displaystyle\qquad\frac{1}{p_{1}}+\frac{1}{p_{2}}$
$\displaystyle=\frac{1}{p_{0}},$
$\displaystyle\quad\frac{1}{q_{1}}+\frac{1}{q_{2}}$
$\displaystyle=\theta_{1}+\frac{1}{q_{0}},$ $\displaystyle
M^{p_{1},q_{1}}*M^{p_{2},q_{2}}$ $\displaystyle\subseteq M^{p_{0},q_{0}},$
$\displaystyle\qquad\frac{1}{p_{1}}+\frac{1}{p_{2}}$
$\displaystyle=\theta_{2}+\frac{1}{p_{0}},$
$\displaystyle\quad\frac{1}{q_{1}}+\frac{1}{q_{2}}$
$\displaystyle=\frac{1}{q_{0}}.$

The general multiplication and convolution properties in Section 2 also
overlap with results by Bastianoni, Cordero and Nicola in [1], by Bastianoni
and Teofanov in [2], and by Guo, Chen, Fan and Zhao in [33].

The multiplication relation in Proposition 0.1 for $p_{j},q_{j}\geq 1$ was
obtained already in [17] by Feichtinger. It is also obvious that the
convolution relation was well-known since then (though a first formal proof of
this relation seems to be given in [49]). In general, these convolution and
multiplication properties follow the rules

$\displaystyle\ell^{p_{1}}$ $\displaystyle*\ell^{p_{2}}\subseteq\ell^{p_{0}},$
$\displaystyle\quad\ell^{q_{1}}$
$\displaystyle\cdot\ell^{q_{2}}\subseteq\ell^{q_{0}}$
$\displaystyle\Rightarrow$ $\displaystyle\quad M^{p_{1},q_{1}}$
$\displaystyle*M^{p_{2},q_{2}}\subseteq M^{p_{0},q_{0}}$ and
$\displaystyle\ell^{p_{1}}$
$\displaystyle\cdot\ell^{p_{2}}\subseteq\ell^{p_{0}},$
$\displaystyle\quad\ell^{q_{1}}$
$\displaystyle*\ell^{q_{2}}\subseteq\ell^{q_{0}}$ $\displaystyle\Rightarrow$
$\displaystyle\quad M^{p_{1},q_{1}}$ $\displaystyle\cdot
M^{p_{2},q_{2}}\subseteq M^{p_{0},q_{0}},$

which goes back to [17] in the Banach space case and to [26] in the quasi-
Banach case. See also [20] and [43] for extensions of these relations to more
general Banach function spaces and quasi-Banach function spaces, respectively.

In Section 2 we basically review some results from [55]. To make this survey
self-contained we give the proof of Theorem 2.7 in unweighted case. In
contrast to [33], we do not deduce any sharpness for our results.

To show Proposition 0.1 in the quasi-Banach setting, apart from the usual use
of of Hölder’s and Young’s inequalities, additional arguments are needed. In
our situation we discretize the situations in similar ways as in [1] by using
Gabor analysis for modulation spaces, and then apply some further arguments,
valid in non-convex analysis. This approach is slightly different compared to
what is used in [33] which follows the discretization technique introduced in
[56], and which has some traces of Gabor analysis.

We refer to [55] for a detailed discussion on the uniqueness of
multiplications and convolutions in Proposition 0.1.

In Section 3 we apply the results from previous parts in the framework of the
so called Gabor product. It is introduced in [14] in order to derive a phase
space analogue to the usual convolution identity for the Fourier transform.
The main motivation is to use such kind of products in a phase-space
formulation of certain nonlinear equations. As noticed in [14], among other
interesting characteristics of phase-space representations, the initial value
problem in phase-space may be well-posed for more general initial
distributions. This means that the phase-space formulation could contain
solutions other than the standard ones. We refer to [11, 12, 13], where the
phase-space extensions are explored in different contexts. Here we illustrate
this approach by considering the nonlinear cubic Schrödinger equation, which
appear for example in in Bose-Einstein condensate theory [36]. We also refer
to [6, Chapter 7] for an overview of results related to well-posedness of the
nonlinear Schrödinger equations in the framework of modulation spaces, see
also [5, 39, 40].

## Acknowledgement

The work of N. Teofanov is partially supported by TIFREFUS Project DS 15, and
MPNTR of Serbia Grant No. 451–03–68/2022–14/200125. Joachim Toft was supported
by Vetenskapsrådet (Swedish Science Council) within the project 2019-04890.

##  1\. Preliminaries

In this section we give an exposition of background material related to the
definition and basic properties of modulation spaces. Thus we recall some
facts on the short-time Fourier transform and related projections, the
(Fourier invariant) Gelfand-Shilov spaces, weight functions, and mixed-norm
spaces of Lebesgue type. We also recall convolution and multiplication in
weighted Lebesgue sequence spaces.

###  1.1. The short-time Fourier transform

In what follows we let $\mathscr{F}$ be the Fourier transform which takes the
form

$(\mathscr{F}f)(\xi)=\widehat{f}(\xi)\equiv(2\pi)^{-\frac{d}{2}}\int_{\mathbf{R}^{d}}f(x)e^{-i\langle
x,\xi\rangle}\,dx$

when $f\in L^{1}(\mathbf{R}^{d})$. Here $\langle\,\cdot\,,\,\cdot\,\rangle$
denotes the usual scalar product on $\mathbf{R}^{d}$. The same notation is
used for the usual dual form between test functions and corresponding
(ultra-)distributions. We recall that map $\mathscr{F}$ extends uniquely to a
homeomorphism on the space of tempered distributions
$\mathscr{S}^{\prime}(\mathbf{R}^{d})$, to a unitary operator on
$L^{2}(\mathbf{R}^{d})$ and restricts to a homeomorphism on the Schwartz space
of smooth rapidly decreasing functions $\mathscr{S}(\mathbf{R}^{d})$, cf.
(1.29). We also observe with our choice of the Fourier transform, the usual
convolution identity for the Fourier transform takes the forms

$\mathscr{F}(f\cdot
g)=(2\pi)^{-\frac{d}{2}}\widehat{f}*\widehat{g}\quad\text{and}\quad\mathscr{F}(f*g)=(2\pi)^{\frac{d}{2}}\widehat{f}\cdot\widehat{g}$
(1.1)

when $f,g\in\mathscr{S}(\mathbf{R}^{d})$.

In several situations it is convenient to use a localized version of the
Fourier transform, called the short-time Fourier transform, STFT for short.
The short-time Fourier transform of $f\in\mathscr{S}^{\prime}(\mathbf{R}^{d})$
with respect to the fixed _window function_
$\phi\in\mathscr{S}(\mathbf{R}^{d})$ is defined by

$\displaystyle(V_{\phi}f)(x,\xi)$
$\displaystyle\equiv(2\pi)^{-\frac{d}{2}}(f,\phi(\,\cdot\,-x)e^{i\langle\,\cdot\,,\xi\rangle})_{L^{2}}.$
(1.2) Here $(\,\cdot\,,\,\cdot\,)_{L^{2}}$ denotes the unique continuous
extension of the inner product on $L^{2}(\mathbf{R}^{d})$ restricted to
$\mathscr{S}(\mathbf{R}^{d})$ into a continuous map from
$\mathscr{S}^{\prime}(\mathbf{R}^{d})\times\mathscr{S}(\mathbf{R}^{d})$ to
$\mathbf{C}$.
We observe that using certain properties for tensor products of distributions,
$\displaystyle(V_{\phi}f)(x,\xi)$
$\displaystyle=\mathscr{F}(f\cdot\overline{\phi(\,\cdot\,-x)})(\xi).$ (1.2)′
(cf. [34, 53]). If in addition $f\in L^{p}(\mathbf{R}^{d})$ for some
$p\in[1,\infty]$, then $\displaystyle(V_{\phi}f)(x,\xi)$
$\displaystyle=(2\pi)^{-\frac{d}{2}}\int_{\mathbf{R}^{d}}f(y)\overline{\phi(y-x)}e^{-i\langle
y,\xi\rangle}\,dy.$ (1.2)′′

We observe that the domain of $V_{\phi}$ is
$\mathscr{S}^{\prime}(\mathbf{R}^{d})$. The images are contained in
$C^{\infty}(\mathbf{R}^{2d})$, the set of smooth functions defined on the
phase space $\mathbf{R}^{d}\times\mathbf{R}^{d}\simeq\mathbf{R}^{2d}$.

The short-time Fourier transform appears in different contexts and under
different names. In quantum mechanics it is rather common to it the _coherent
state transform_ (see e. g. [38]). It is also closely related to the so-called
Wigner distribution or radar ambiguity function (see e. g. [37]). In time-
frequency analysis, it is also sometimes called a _Voice transform_.

The main idea with the design of short-time Fourier transform is to get the
Fourier content, or the frequency resolution of localized functions and
distributions. Roughly speaking, short-time Fourier transform give a
simultaneous information both on functions or distributions themselves as well
as their Fourier transforms in the sense that the map

$x\mapsto V_{\phi}f(x,\xi)$

resembles on $f(x)$, while the map

$\xi\mapsto V_{\phi}f(x,\xi)$

resembles on $\widehat{f}(\xi)$.

As for the ordinary Fourier transform, there are several mapping properties
which hold true for the short-time Fourier transform. An elegant way to
approach such mapping in the framework of distributions, we may follow ideas
given in [25] by Folland.

In fact, let $T$ is the semi-conjugated tensor map

$\displaystyle T(f,\phi)$ $\displaystyle=f\otimes\overline{\phi},$ (1.3) $U$
be the linear pullback $\displaystyle(UF)(x,y)$ $\displaystyle=U(y,y-x)$ (1.4)
and $\mathscr{F}_{2}$ be the partial Fourier transform given by
$\displaystyle(\mathscr{F}_{2}F)(x,\xi)$
$\displaystyle=(2\pi)^{-\frac{d}{2}}\int_{\mathbf{R}^{d}}F(x,y)e^{-i\langle
y,\xi\rangle}\,dy.$ (1.5)

Then

$V_{\phi}f=(\mathscr{F}_{2}\circ U\circ T)(f,\phi),$ (1.6)

when $f,\phi\in\mathscr{S}(\mathbf{R}^{d})$.

We observe that the mappings

$\displaystyle T:\mathscr{S}(\mathbf{R}^{d})\times\mathscr{S}(\mathbf{R}^{d})$
$\displaystyle\to\mathscr{S}(\mathbf{R}^{2d}),$ $\displaystyle\quad
U,\mathscr{F}_{2}:\mathscr{S}(\mathbf{R}^{2d})$
$\displaystyle\to\mathscr{S}(\mathbf{R}^{2d})$ (1.7) are continuous and
uniquely extendable to continuous mappings $\displaystyle
T:\mathscr{S}^{\prime}(\mathbf{R}^{d})\times\mathscr{S}^{\prime}(\mathbf{R}^{d})$
$\displaystyle\to\mathscr{S}^{\prime}(\mathbf{R}^{2d}),$ $\displaystyle\quad
U,\mathscr{F}_{2}:\mathscr{S}^{\prime}(\mathbf{R}^{2d})$
$\displaystyle\to\mathscr{S}^{\prime}(\mathbf{R}^{2d}),$ (1.8) which in turn
restricts to isometric mappings $\displaystyle T:L^{2}(\mathbf{R}^{d})\times
L^{2}(\mathbf{R}^{d})$ $\displaystyle\to L^{2}(\mathbf{R}^{2d}),$
$\displaystyle\quad U,\mathscr{F}_{2}:L^{2}(\mathbf{R}^{2d})$
$\displaystyle\to L^{2}(\mathbf{R}^{2d}).$ (1.9)

Here that $T$ is isometric means that

$\|T(f,\phi)\|_{L^{2}(\mathbf{R}^{2d})}=\|f\|_{L^{2}(\mathbf{R}^{d})}\|\phi\|_{L^{2}(\mathbf{R}^{d})}.$

It is now natural to define $V_{\phi}f$ as the right-hand side of (1.6) when
$f,\phi\in\mathscr{S}^{\prime}(\mathbf{R}^{d})$, in which $V_{\phi}f$ is well-
defined as an element in $\mathscr{S}^{\prime}(\mathbf{R}^{2d})$.

######  Proposition 1.1.

The map

$\displaystyle(f,\phi)$ $\displaystyle\mapsto V_{\phi}f$ $\displaystyle:\,$
$\displaystyle\mathscr{S}(\mathbf{R}^{d})\times\mathscr{S}(\mathbf{R}^{d})$
$\displaystyle\to\mathscr{S}(\mathbf{R}^{2d})$ (1.10) is continuous, which
extends uniquely to a continuous map $\displaystyle(f,\phi)$
$\displaystyle\mapsto V_{\phi}f$ $\displaystyle:\,$
$\displaystyle\mathscr{S}^{\prime}(\mathbf{R}^{d})\times\mathscr{S}^{\prime}(\mathbf{R}^{d})$
$\displaystyle\to\mathscr{S}^{\prime}(\mathbf{R}^{2d}),$ (1.11) which in turn
restricts to an isometric map $\displaystyle(f,\phi)$ $\displaystyle\mapsto
V_{\phi}f$ $\displaystyle:\,$ $\displaystyle L^{2}(\mathbf{R}^{d})\times
L^{2}(\mathbf{R}^{d})$ $\displaystyle\to L^{2}(\mathbf{R}^{2d}).$ (1.12)

If $\phi\in\mathscr{S}(\mathbf{R}^{d})$ and
$f\in\mathscr{S}^{\prime}(\mathbf{R}^{d})$, then (1.11) shows that
$V_{\phi}f\in\mathscr{S}^{\prime}(\mathbf{R}^{2d})$. On the other hand, it is
easy to see that the right-hand side of (1.2) defines a smooth function.
Consequently beside (1.11) and (1.10), we also have the continuous map

$(f,\phi)\mapsto
V_{\phi}f:\,\mathscr{S}^{\prime}(\mathbf{R}^{d})\times\mathscr{S}(\mathbf{R}^{d})\to\mathscr{S}^{\prime}(\mathbf{R}^{2d})\cap
C^{\infty}(\mathbf{R}^{2d}).$ (1.13)

For short-time Fourier transform, the Parseval identity is replaced by the so-
called Moyal identity, also known as the orthogonality relation given by

$(V_{\phi}f,V_{\psi}g)_{L^{2}(\mathbf{R}^{2d})}=(\psi,\phi)_{L^{2}(\mathbf{R}^{d})}(f,g)_{L^{2}(\mathbf{R}^{d})},$
(1.14)

when $f,g,\phi,\psi\in\mathscr{S}(\mathbf{R}^{d})$. The identity (1.14) is
obtained by rewriting the short-time Fourier transforms by (1.2)′ and then
applying the Parseval identity in suitable ways. We observe that the right-
hand side makes sense also when $f$, $g$, $\phi$ and $\psi$ belong to other
spaces than $\mathscr{S}(\mathbf{R}^{d})$. For example we may let

$\displaystyle(f,g,\phi,\psi)$
$\displaystyle\in\mathscr{S}^{\prime}(\mathbf{R}^{d})\times\mathscr{S}(\mathbf{R}^{d})\times\mathscr{S}(\mathbf{R}^{d})\times\mathscr{S}^{\prime}(\mathbf{R}^{d}),$
(1.15) $\displaystyle(f,g,\phi,\psi)$
$\displaystyle\in\mathscr{S}(\mathbf{R}^{d})\times\mathscr{S}^{\prime}(\mathbf{R}^{d})\times\mathscr{S}^{\prime}(\mathbf{R}^{d})\times\mathscr{S}(\mathbf{R}^{d}),$
$\displaystyle(f,g,\phi,\psi)$
$\displaystyle\in\mathscr{S}^{\prime}(\mathbf{R}^{d})\times\mathscr{S}(\mathbf{R}^{d})\times
L^{q}(\mathbf{R}^{d})\times L^{q^{\prime}}(\mathbf{R}^{d})$
$\displaystyle\text{or}\qquad(f,g,\phi,\psi)$ $\displaystyle\in
L^{p}(\mathbf{R}^{d})\times L^{p^{\prime}}(\mathbf{R}^{d})\times
L^{q}(\mathbf{R}^{d})\times L^{q^{\prime}}(\mathbf{R}^{d}),$

when $p,p^{\prime},q,q^{\prime}\in[1,\infty]$ satisfy

$\frac{1}{p}+\frac{1}{p^{\prime}}=\frac{1}{q}+\frac{1}{q^{\prime}}=1.$

By Moyal’s identity (1.14) it follows that if
$\phi\in\mathscr{S}(\mathbf{R}^{d})\setminus 0$, then the identity operator on
$\mathscr{S}^{\prime}(\mathbf{R}^{d})$ is given by

$\operatorname{Id}=\left(\|\phi\|_{L^{2}}^{-2}\right)\cdot V_{\phi}^{*}\circ
V_{\phi},$ (1.16)

provided suitable mapping properties of the ($L^{2}$-)adjoint $V_{\phi}^{*}$
of $V_{\phi}$ can be established. Obviously, $V_{\phi}^{*}$ fullfils

$(V_{\phi}^{*}F,g)_{L^{2}(\mathbf{R}^{d})}=(F,V_{\phi}g)_{L^{2}(\mathbf{R}^{2d})}$
(1.17)

when $F\in\mathscr{S}(\mathbf{R}^{2d})$ and $g\in\mathscr{S}(\mathbf{R}^{d})$.

By expressing the scalar product and the short-time Fourier transform in terms
of integrals in (1.17), it follows by straight-forward manipulations that the
adjoint in (1.17) is given by

$(V_{\phi}^{*}F)(x)=(2\pi)^{-\frac{d}{2}}\iint_{\mathbf{R}^{2d}}F(y,\eta)\phi(x-y)e^{i\langle
x,\eta\rangle}\,dyd\eta,$ (1.18)

when $F\in\mathscr{S}(\mathbf{R}^{2d})$. We may now use mapping properties
like (1.11)–(1.12) to extend the definition of $V_{\phi}^{*}F$ when $F$ and
$\phi$ belong to various classes of function and distribution spaces. For
example, by (1.11), (1.10) and (1.12), it follows that the map

$(F,g)\mapsto(F,V_{\phi}g)_{L^{2}(\mathbf{R}^{2d})}$

defines a sesqui-linear form on
$\mathscr{S}(\mathbf{R}^{2d})\times\mathscr{S}^{\prime}(\mathbf{R}^{d})$,
$\mathscr{S}^{\prime}(\mathbf{R}^{2d})\times\mathscr{S}(\mathbf{R}^{d})$ and
on $L^{2}(\mathbf{R}^{2d})\times L^{2}(\mathbf{R}^{d})$. This implies that if
$\phi\in\mathscr{S}(\mathbf{R}^{d})$, then $V_{\phi}^{*}$ in (1.17) is
continuous from $\mathscr{S}(\mathbf{R}^{2d})$ to
$\mathscr{S}(\mathbf{R}^{d})$ which is uniquely extendable to a continuous map
$\mathscr{S}^{\prime}(\mathbf{R}^{2d})$ to
$\mathscr{S}^{\prime}(\mathbf{R}^{d})$, and to $L^{2}(\mathbf{R}^{2d})$ to
$L^{2}(\mathbf{R}^{d})$. That is, the mappings

$\displaystyle
V_{\phi}^{*}:\mathscr{S}(\mathbf{R}^{2d})\to\mathscr{S}(\mathbf{R}^{d}),$
$\displaystyle V_{\phi}^{*}$ $\displaystyle:$
$\displaystyle\mathscr{S}^{\prime}(\mathbf{R}^{2d})$
$\displaystyle\to\mathscr{S}^{\prime}(\mathbf{R}^{d})$ (1.19) and
$\displaystyle V_{\phi}^{*}$ $\displaystyle:$ $\displaystyle
L^{2}(\mathbf{R}^{2d})$ $\displaystyle\to L^{2}(\mathbf{R}^{d})$

are continuous.

###  1.2. STFT projections and a suitable twisted convolution

If $\phi\in\mathscr{S}(\mathbf{R}^{d})$ satisfies $\|\phi\|_{L^{2}}=1$, then
(1.16) shows that $V_{\phi}^{*}\circ V_{\phi}$ is the identity operator on
$\mathscr{S}^{\prime}(\mathbf{R}^{d})$. If we swap the order of this
composition we get certain types of projections. In fact, for any
$\phi\in\mathscr{S}(\mathbf{R}^{d})\setminus 0$, let $P_{\phi}$ be the
operator given by

$P_{\phi}\equiv\|\phi\|_{L^{2}}^{-2}\cdot V_{\phi}\circ V_{\phi}^{*}.$ (1.20)

We observe that $P_{\phi}$ is continuous on $\mathscr{S}(\mathbf{R}^{2d})$,
$L^{2}(\mathbf{R}^{2d})$ and $\mathscr{S}^{\prime}(\mathbf{R}^{2d})$ due to
the mapping properties for $V_{\phi}$ and $V_{\phi}^{*}$ above.

It is clear that $P_{\phi}^{*}=P_{\phi}$, i. e. $P_{\phi}$ is self-adjoint.
Furthermore, $P_{\phi}$ is an involution:

$P_{\phi}^{2}=\|\phi\|_{L^{2}}^{-2}\cdot
V_{\phi}\circ\Big{(}\underset{\text{The identity
operator}}{\underbrace{\|\phi\|_{L^{2}}^{-2}\cdot V_{\phi}^{*}\circ
V_{\phi}}}\Big{)}\circ V_{\phi}^{*}=\|\phi\|_{L^{2}}^{-2}\cdot V_{\phi}\circ
V_{\phi}^{*}=P_{\phi}.$

Hence,

$P_{\phi}^{*}=P_{\phi}\quad\text{and}\quad P_{\phi}^{2}=P_{\phi},$ (1.21)

which shows that $P_{\phi}$ is an orthonormal projection.

The ranks of $P_{\phi}$ are given by

$\displaystyle
P_{\phi}(\mathscr{S}(\mathbf{R}^{2d}))=V_{\phi}(\mathscr{S}(\mathbf{R}^{d})),\qquad
P_{\phi}(L^{2}(\mathbf{R}^{2d}))$
$\displaystyle=V_{\phi}(L^{2}(\mathbf{R}^{d})),$ (1.22)
$\displaystyle\text{and}\qquad
P_{\phi}(\mathscr{S}^{\prime}(\mathbf{R}^{2d}))$
$\displaystyle=V_{\phi}(\mathscr{S}^{\prime}(\mathbf{R}^{d})).$

In fact, if $F\in\mathscr{S}^{\prime}(\mathbf{R}^{2d})$, then

$P_{\phi}F=V_{\phi}f,$

where
$f=\|\phi\|_{L^{2}}^{-2}V_{\phi}^{*}F\in\mathscr{S}^{\prime}(\mathbf{R}^{d})$.
This shows that $P_{\phi}(\mathscr{S}^{\prime}(\mathbf{R}^{2d}))\subseteq
V_{\phi}(\mathscr{S}^{\prime}(\mathbf{R}^{d}))$. On the other hand, if
$f\in\mathscr{S}^{\prime}(\mathbf{R}^{d})$ and $F=V_{\phi}f$, then

$P_{\phi}F=\Big{(}V_{\phi}\circ\Big{(}\|\phi\|_{L^{2}}^{-2}\cdot
V_{\phi}^{*}\circ V_{\phi}\Big{)}\Big{)}f=V_{\phi}f,$

which shows that any element in
$V_{\phi}(\mathscr{S}^{\prime}(\mathbf{R}^{d}))$ equals to an element in
$P_{\phi}(\mathscr{S}^{\prime}(\mathbf{R}^{2d}))$, i.e.
$P_{\phi}(\mathscr{S}^{\prime}(\mathbf{R}^{2d}))=V_{\phi}(\mathscr{S}^{\prime}(\mathbf{R}^{d}))$.
This gives the last identity in (1.22). In the same way, the first two
identities are obtained.

######  Remark 1.2.

Let $F\in\mathscr{S}^{\prime}(\mathbf{R}^{2d})$. Then it follows from the last
identity in (1.22) that $F=V_{\phi}f$ for some
$f\in\mathscr{S}^{\prime}(\mathbf{R}^{d})$, if and only if

$F=P_{\phi}F.$ (1.23)

Furthermore, if (1.23) holds, then $F=V_{\phi}f$ with

$f=(\|\phi\|_{L^{2}}^{-2})\cdot V_{\phi}^{*}F.$ (1.24)

There is a twisted convolution which is linked to the projection in (1.20). In
fact, if $F\in\mathscr{S}(\mathbf{R}^{2d})$ and
$\phi\in\mathscr{S}(\mathbf{R}^{d})\setminus 0$, then it follows by expanding
the integrals for $V_{\phi}$ and $V_{\phi}^{*}$ in (1.20), and performing some
straight-forward manipulations that

$P_{\phi}F=\|\phi\|_{L^{2}}^{-2}\cdot V_{\phi}\phi*_{V}F,\qquad
F\in\mathscr{S}^{\prime}(\mathbf{R}^{2d}),$ (1.25)

where the _twisted convolution_ $*_{V}$ is defined by

$\displaystyle(F*_{V}G)(x,\xi)$
$\displaystyle=(2\pi)^{-\frac{d}{2}}\iint_{\mathbf{R}^{2d}}\\!\\!F(x-y,\xi-\eta)G(y,\eta)e^{-i\langle
y,\xi-\eta\rangle}\,dyd\eta.$
$\displaystyle=(2\pi)^{-\frac{d}{2}}\iint_{\mathbf{R}^{2d}}\\!\\!F(y,\eta)G(x-y,\xi-\eta)e^{-i\langle
x-y,\eta\rangle}\,dyd\eta,$ (1.26)

when $F,G\in\mathscr{S}(\mathbf{R}^{2d})$. We observe that the definition of
$*_{V}$ is uniquely extendable in different ways. For example, Young’s
inequality for ordinary convolution also holds for the twisted convolution.
Moreover, the map $(F,G)\mapsto F*_{V}G$ extends uniquely to continuous
mappings from
$\mathscr{S}(\mathbf{R}^{2d})\times\mathscr{S}^{\prime}(\mathbf{R}^{2d})$ or
$\mathscr{S}^{\prime}(\mathbf{R}^{2d})\times\mathscr{S}(\mathbf{R}^{2d})$ to
$\mathscr{S}^{\prime}(\mathbf{R}^{2d})$. By straight-forward computations it
follows that

$(F*_{V}G)*_{V}H=F*_{V}(G*_{V}H),$ (1.27)

when $F,H\in\mathscr{S}(\mathbf{R}^{2d})$ and
$G\in\mathscr{S}^{\prime}(\mathbf{R}^{2d})$, or
$F,H\in\mathscr{S}^{\prime}(\mathbf{R}^{2d})$ and
$G\in\mathscr{S}(\mathbf{R}^{2d})$

Let $f\in\mathscr{S}^{\prime}(\mathbf{R}^{d})$ and
$\phi_{j}\in\mathscr{S}(\mathbf{R}^{d})$, $j=1,2,3$. By straight-forward
applications of Parseval’s formula it follows that

$\big{(}(V_{\phi_{2}}\phi_{3})*_{V}(V_{\phi_{1}}f)\big{)}(x,\xi)=(\phi_{3},\phi_{1})_{L^{2}}\cdot(V_{\phi_{2}}f)(x,\xi),$
(1.28)

which is some sort of reproducing kernel of short-time Fourier transforms in
the background of $*_{V}$.

###  1.3. Gelfand-Shilov spaces

Before defining the Gelfand-Shilov spaces, we recall that the Schwartz space
$\mathscr{S}(\mathbf{R}^{d})$ consists of all (complex-valued) smooth
functions $f\in C^{\infty}(\mathbf{R}^{d})$ such that

$\sup_{x\in\mathbf{R}^{d}}\big{(}|x^{\beta}\partial^{\alpha}f(x)|\big{)}\leq
C_{\alpha,\beta},$ (1.29)

for some constants $C_{\alpha,\beta}>0$, which only depend on the multi-
indices $\alpha,\beta\in\mathbf{N}^{d}$. The Schwartz space possess several
convenient properties. and is heavily used in mathematics, science and
technology. For example, the Schwartz space is invariant under Fourier
transformation. By duality the same holds true for its ($L^{2}$-)dual
$\mathscr{S}^{\prime}(\mathbf{R}^{d})$, the set of tempered distributions on
$\mathbf{R}^{d}$.

On the other hand, we observe that there are no conditions on the growths of
the constants $C_{\alpha,\beta}$ with respect to
$\alpha,\beta\in\mathbf{N}^{d}$. This implies that in the context of the
spaces $\mathscr{S}(\mathbf{R}^{d})$ and
$\mathscr{S}^{\prime}(\mathbf{R}^{d})$, it is almost impossible to investigate
important properties like analyticity or related regularity properties which
are stronger than pure smoothness. In order for investigating such stronger
regularity properties, we need to modify $\mathscr{S}(\mathbf{R}^{d})$ and the
estimate (1.29) by imposing suitable growth conditions on the constants
$C_{\alpha,\beta}$. This leads to the definition of Gelfand-Shilov spaces,
[27, 41].

We only discuss Fourier invariant Gelfand-Shilov spaces and their properties.
Let $0<s\in\mathbf{R}$ be fixed. We have two different types of Gelfand-Shilov
spaces. The Gelfand-Shilov space $\mathcal{S}_{s}(\mathbf{R}^{d})$ of Roumieu
type with parameter $s$ consists of all $f\in C^{\infty}(\mathbf{R}^{d})$ such
that

$\sup_{x\in\mathbf{R}^{d}}\big{(}|x^{\beta}\partial^{\alpha}f(x)|\big{)}\leq
Ch^{|\alpha+\beta|}(\alpha!\beta!)^{s},$ (1.30)

for some constants $C,h>0$. In the same way, the Gelfand-Shilov space
$\Sigma_{s}(\mathbf{R}^{d})$ of Beurling type with parameter $s$ consists of
all $f\in C^{\infty}(\mathbf{R}^{d})$ such that for every $h>0$, there is a
constant $C=C_{h}>0$ such that (1.30). Hence, in comparison with the
definition of Schwartz functions, we have limited ourself to constants
$C_{\alpha,\beta}$ in (1.29) which are not allowed to grow faster than those
of the form

$Ch^{|\alpha+\beta|}(\alpha!\beta!)^{s}$

when dealing with Gelfand-Shilov spaces.

It can be proved that $\mathcal{S}_{s}(\mathbf{R}^{d})$ and
$\Sigma_{t}(\mathbf{R}^{d})$ are dense in $\mathscr{S}(\mathbf{R}^{d})$ when
$s\geq\frac{1}{2}$ and $t>\frac{1}{2}$. We call such $s$ and $t$ admissible.
On the other hand, for the other choices of $s$ and $t$ we have

$\mathcal{S}_{s}(\mathbf{R}^{d})=\Sigma_{t}(\mathbf{R}^{d})=\\{0\\},\quad\text{when}\quad
s<\frac{1}{2},\ t\leq\frac{1}{2}.$

One has that $\mathcal{S}_{1}(\mathbf{R}^{d})$ consists of real analytic
functions, and that $\Sigma_{1}(\mathbf{R}^{d})$ consists of smooth functions
on $\mathbf{R}^{d}$ which are extendable to entire functions on
$\mathbf{C}^{d}$. The topologies of $\mathcal{S}_{s}(\mathbf{R}^{d})$ and
$\Sigma_{s}(\mathbf{R}^{d})$ are defined by the semi-norms

$\|f\|_{\mathcal{S}_{s,h}}\equiv\sup\frac{|x^{\beta}\partial^{\alpha}f(x)|}{h^{|\alpha+\beta|}(\alpha!\beta!)^{s}}.$
(1.31)

Here the supremum should be taken over all $\alpha,\beta\in\mathbf{N}^{d}$ and
$x\in\mathbf{R}^{d}$. We equip $\mathcal{S}_{s}(\mathbf{R}^{d})$ and
$\Sigma_{s}(\mathbf{R}^{d})$ by the canonical inductive limit topology and
projective limit topology, respectively, with respect to $h>0$, which are
induced by the semi-norms in (1.31).

Let $\mathcal{S}_{s,h}(\mathbf{R}^{d})$ be the Banach space which consists of
all $f\in C^{\infty}(\mathbf{R}^{d})$ such that $\|f\|_{\mathcal{S}_{s,h}}$ in
(1.31) is finite, and let $\mathcal{S}_{s,h}^{\prime}(\mathbf{R}^{d})$ be the
($L^{2}$-)dual of $\mathcal{S}_{s,h}(\mathbf{R}^{d})$. If $s\geq\frac{1}{2}$,
then the _Gelfand-Shilov distribution space_
$\mathcal{S}_{s}^{\prime}(\mathbf{R}^{d})$ of _Roumieu type_ is the projective
limit of $\mathcal{S}_{s,h}^{\prime}(\mathbf{R}^{d})$ with respect to $h>0$.
If instead $s>\frac{1}{2}$, then the _Gelfand-Shilov distribution space_
$\Sigma_{s}^{\prime}(\mathbf{R}^{d})$ of _Beurling type_ is the inductive
limit of $\mathcal{S}_{s,h}^{\prime}(\mathbf{R}^{d})$ with respect to $h>0$.
Consequently, for admissible $s$ we have

$\mathcal{S}_{s}^{\prime}(\mathbf{R}^{d})=\bigcap_{h>0}\mathcal{S}_{s,h}^{\prime}(\mathbf{R}^{d})\quad\text{and}\quad\Sigma_{s}^{\prime}(\mathbf{R}^{d})=\bigcup_{h>0}\mathcal{S}_{s,h}^{\prime}(\mathbf{R}^{d}).$

It can be proved that $\mathcal{S}_{s}^{\prime}(\mathbf{R}^{d})$ and
$\Sigma_{s}^{\prime}(\mathbf{R}^{d})$ are the (strong) duals to
$\mathcal{S}_{s}(\mathbf{R}^{d})$ and $\Sigma_{s}(\mathbf{R}^{d})$,
respectively.

We have the following embeddings and density properties for Gelfand-Shilov and
Schwartz spaces we get

$\displaystyle\mathcal{S}_{s}(\mathbf{R}^{d})$ $\displaystyle\hookrightarrow$
$\displaystyle\Sigma_{t}(\mathbf{R}^{d})$ $\displaystyle\hookrightarrow$
$\displaystyle\mathcal{S}_{t}(\mathbf{R}^{d})$
$\displaystyle\hookrightarrow\mathscr{S}(\mathbf{R}^{d}),$ (1.32)
$\displaystyle\mathscr{S}^{\prime}(\mathbf{R}^{d})$
$\displaystyle\hookrightarrow$
$\displaystyle\mathcal{S}_{t}^{\prime}(\mathbf{R}^{d})$
$\displaystyle\hookrightarrow$
$\displaystyle\Sigma_{t}^{\prime}(\mathbf{R}^{d})$
$\displaystyle\hookrightarrow\mathcal{S}_{s}^{\prime}(\mathbf{R}^{d}),\qquad
t>s\geq\frac{1}{2},$

with dense embeddings. Here $A\hookrightarrow B$ means that the topological
spaces $A$ and $B$ satisfy $A\subseteq B$ with continuous embeddings.

The Fourier transform possess convenient mapping properties on Gelfand-Shilov
spaces and their distribution spaces. In fact, the Fourier transform extends
uniquely to homeomorphisms on $\mathscr{S}^{\prime}(\mathbf{R}^{d})$,
$\mathcal{S}_{s}^{\prime}(\mathbf{R}^{d})$ and on
$\Sigma_{s}^{\prime}(\mathbf{R}^{d})$ for admissible $s$. Furthermore,
$\mathscr{F}$ restricts to homeomorphisms on $\mathcal{S}_{s}(\mathbf{R}^{d})$
and on $\Sigma_{s}(\mathbf{R}^{d})$.

One of the most important characterizations of Gelfand-Shilov spaces is
performed in terms of estimates of the functions and their Fourier transforms.
More precisely, in [8, 15] it is proved that if
$f\in\mathscr{S}^{\prime}(\mathbf{R}^{d})$ and $s>0$, then
$f\in\mathcal{S}_{s}(\mathbf{R}^{d})$ ($f\in\Sigma_{s}(\mathbf{R}^{d})$), if
and only if

$|f(x)|\lesssim
e^{-r|x|^{\frac{1}{s}}}\quad\text{and}\quad|\widehat{f}(\xi)|\lesssim
e^{-r|\xi|^{\frac{1}{s}}},$ (1.33)

for some $r>0$ (for every $r>0$). Here $g_{1}\lesssim g_{2}$ means that
$g_{1}(\theta)\leq c\cdot g_{2}(\theta)$ holds uniformly for all $\theta$ in
the intersection of the domains of $g_{1}$ and $g_{2}$ for some constant
$c>0$, and we write $g_{1}\asymp g_{2}$ when $g_{1}\lesssim g_{2}\lesssim
g_{1}$.

The analysis in [8, 15] can also be applied on the Schwartz space, from which
it follows that an element $f\in\mathscr{S}^{\prime}(\mathbf{R}^{d})$ belongs
to $\mathscr{S}(\mathbf{R}^{d})$, if and only if

$|f(x)|\lesssim\langle
x\rangle^{-N}\quad\text{and}\quad|\widehat{f}(\xi)|\lesssim\langle\xi\rangle^{-N},$
(1.34)

for every $N\geq 0$. Here and in what follows we let

$\langle x\rangle=(1+|x|^{2})^{\frac{1}{2}}.$

######  Remark 1.3.

Several properties in Subsections 1.1 – 1.3 in the background of
$\mathscr{S}(\mathbf{R}^{d})$ and $\mathscr{S}^{\prime}(\mathbf{R}^{d})$ also
hold for the Gelfand-Shilov spaces and their distribution spaces. Let
$s\geq\frac{1}{2}$. By similar arguments which lead to Proposition 1.1 and
(1.13), it follows that

$\displaystyle(f,\phi)$ $\displaystyle\mapsto V_{\phi}f$ $\displaystyle:\,$
$\displaystyle\mathcal{S}_{s}(\mathbf{R}^{d})\times\mathcal{S}_{s}(\mathbf{R}^{d})$
$\displaystyle\to\mathcal{S}_{s}(\mathbf{R}^{2d})$ (1.35) is continuous, which
extends uniquely to continuous mappings $\displaystyle(f,\phi)$
$\displaystyle\mapsto V_{\phi}f$ $\displaystyle:\,$
$\displaystyle\mathcal{S}_{s}^{\prime}(\mathbf{R}^{d})\times\mathcal{S}_{s}(\mathbf{R}^{d})$
$\displaystyle\to\mathcal{S}_{s}^{\prime}(\mathbf{R}^{2d})\cap
C^{\infty}(\mathbf{R}^{2d})$ (1.36) and $\displaystyle(f,\phi)$
$\displaystyle\mapsto V_{\phi}f$ $\displaystyle:\,$
$\displaystyle\mathcal{S}_{s}^{\prime}(\mathbf{R}^{d})\times\mathcal{S}_{s}^{\prime}(\mathbf{R}^{d})$
$\displaystyle\to\mathcal{S}_{s}^{\prime}(\mathbf{R}^{2d}).$ (1.37)

It follows that (1.14) makes sense after each $\mathscr{S}$ in (1.15) are
replaced by $\mathcal{S}_{s}$. Let
$\phi\in\mathcal{S}_{s}(\mathbf{R}^{d})\setminus 0$ be fixed. Then by similar
arguments which lead to (1.19) give that the mappings

$V_{\phi}^{*}:\mathcal{S}_{s}(\mathbf{R}^{2d})\to\mathcal{S}_{s}(\mathbf{R}^{d}),\qquad
V_{\phi}^{*}:\mathcal{S}_{s}^{\prime}(\mathbf{R}^{2d})\to\mathcal{S}_{s}^{\prime}(\mathbf{R}^{d})$
(1.19)′

are continuous. For $P_{\phi}$ in (1.20) we have that (1.21) still holds true
and that (1.22) can be completed with

$P_{\phi}(\mathcal{S}_{s}(\mathbf{R}^{2d}))=V_{\phi}(\mathcal{S}_{s}(\mathbf{R}^{d}))\quad\text{and}\quad
P_{\phi}(\mathcal{S}_{s}^{\prime}(\mathbf{R}^{2d}))=V_{\phi}(\mathcal{S}_{s}^{\prime}(\mathbf{R}^{d})).$
(1.38)

We also have that the twisted convolution in (1.26) is continuous from
$\mathcal{S}_{s}(\mathbf{R}^{2d})\times\mathcal{S}_{s}(\mathbf{R}^{2d})$ to
$\mathcal{S}_{s}(\mathbf{R}^{2d})$ and uniquely extendable to a continuous map
$\mathcal{S}_{s}(\mathbf{R}^{2d})\times\mathcal{S}_{s}^{\prime}(\mathbf{R}^{2d})$
or
$\mathcal{S}_{s}^{\prime}(\mathbf{R}^{2d})\times\mathcal{S}_{s}(\mathbf{R}^{2d})$
to $\mathcal{S}_{s}^{\prime}(\mathbf{R}^{2d})$, and that the formulae
(1.25)–(1.28) still hold true after each $\mathscr{S}$ is replaced by
$\mathcal{S}_{s}$ in the attached assumptions.

If instead $s>\frac{1}{2}$, then similar facts hold true with $\Sigma_{s}$ in
place of $\mathcal{S}_{s}$ above, at each occurrence.

######  Remark 1.4.

In similar ways as characterizing Gelfand-Shilov spaces in terms of Fourier
estimates (see (1.33)), we may also use the short-time Fourier transform to
perform similar characterizations. Moreover, the short-time Fourier transform
can in addition be used to characterize spaces of Gelfand-Shilov
distributions.

In fact, let $\phi\in\mathcal{S}_{s}(\mathbf{R}^{d})\setminus 0$
($\phi\in\Sigma_{s}(\mathbf{R}^{d})\setminus 0$) be fixed and let $f$ be a
Gelfand-Shilov distribution on $\mathbf{R}^{d}$. Then the following is true:

  1. (1)

$f\in\mathcal{S}_{s}(\mathbf{R}^{d})$ ($f\in\Sigma_{s}(\mathbf{R}^{d})$), if
and only if

$|V_{\phi}f(x,\xi)|\lesssim e^{-r(|x|^{\frac{1}{s}}+|\xi|^{\frac{1}{s}})}$
(1.39)

for some $r>0$ (for every $r>0$);

  2. (2)

$f\in\mathcal{S}_{s}^{\prime}(\mathbf{R}^{d})$
($f\in\Sigma_{s}^{\prime}(\mathbf{R}^{d})$), if and only if

$|V_{\phi}f(x,\xi)|\lesssim e^{r(|x|^{\frac{1}{s}}+|\xi|^{\frac{1}{s}})}$
(1.40)

for every $r>0$ (for some $r>0$).

We refer to [32, Theorem 2.7] for the characterization (1) concerning Gelfand-
Shilov functions and to [52, Proposition 2.2]) for the characterization (2)
concerning Gelfand-Shilov distributions.

###  1.4. Weight functions

A _weight_ or _weight function_ on $\mathbf{R}^{d}$ is a positive function
$\omega\in L^{\infty}_{loc}(\mathbf{R}^{d})$ such that $1/\omega\in
L^{\infty}_{loc}(\mathbf{R}^{d})$. The weight $\omega$ is called _moderate_ ,
if there is a positive weight $v$ on $\mathbf{R}^{d}$ and a constant $C\geq 1$
such that

$\omega(x+y)\leq C\omega(x)v(y),\qquad x,y\in\mathbf{R}^{d}.$ (1.41)

If $\omega$ and $v$ are weights on $\mathbf{R}^{d}$ such that (1.41) holds,
then $\omega$ is also called _$v$ -moderate_. We note that (1.41) implies that
$\omega$ fulfills the estimates

$C^{-1}v(-x)^{-1}\leq\omega(x)\leq Cv(x),\quad x\in\mathbf{R}^{d}.$ (1.42)

We let $\mathscr{P}_{E}(\mathbf{R}^{d})$ be the set of all moderate weights on
$\mathbf{R}^{d}$.

We say that $v$ is _submultiplicative_ if

$v(x+y)\leq v(x)v(y)\quad\text{and}\quad v(-x)=v(x),\qquad
x,y\in\mathbf{R}^{d}.$ (1.43)

We observe that if $v\in\mathscr{P}_{E}(\mathbf{R}^{d})$ is even and satisfies

$v(x+y)\leq Cv(x)v(y),\qquad x,y\in\mathbf{R}^{d},$ (1.44)

for some constant $C>0$, then for $v_{0}=C^{1/2}v$, one has that
$v_{0}\in\mathscr{P}_{E}(\mathbf{R}^{d})$ is submultiplicative and $v\asymp
v_{0}$ (see e. g. [18, 20, 29]).

We also recall from [30] that if $v$ is positive and locally bounded and
satisfies (1.44), then $v(x)\leq C_{0}e^{r_{0}|x|}$ for some positive
constants $C_{0}$ and $r_{0}$. In fact, if $x\in\mathbf{R}^{d}$,

$r=\sup_{|x|\leq 1}\log v(x),\quad c=\log C$

and $n$ is an integer such that $n-1\leq|x|\leq n$, then (1.44) gives

$v(x)=v(n\cdot(x/n))\leq C^{n}v(x/n)^{n}\leq C^{n}e^{rn}=e^{(r+c)n}\leq
e^{(r+c)(|x|+1)},$

which gives the statement.

Therefore, if $v$ is a submultiplicative weight, then

$v(x)\lesssim e^{r|x|},\qquad x\in\mathbf{R}^{d},$ (1.45)

for some $r\geq 0$. Hence, if $\omega\in\mathscr{P}_{E}(\mathbf{R}^{d})$, then
(1.41) and (1.45) imply

$\omega(x+y)\lesssim\omega(x)e^{r|y|},\quad x,y\in\mathbf{R}^{d}$ (1.46)

for some $r>0$. In particular, (1.42) shows that for any
$\omega_{0}\in\mathscr{P}_{E}(\mathbf{R}^{d})$, there is a constant $r>0$ such
that

$e^{-r|x|}\lesssim\omega_{0}(x)\lesssim e^{r|x|},\quad x\in\mathbf{R}^{d}.$

If (1.41) holds, then there is a smallest positive even function $v_{0}$ such
that (1.41) holds with $C=1$. We remark that this $v_{0}$ is given by

$v_{0}(x)=\sup_{y\in\mathbf{R}^{d}}\left(\frac{\omega(x+y)}{\omega(y)},\frac{\omega(-x+y)}{\omega(y)}\right),$

and is submultiplicative (see e. g. [20, 28, 50]). Consequently, if $\omega$
is a moderate weight, then it is also moderated by submultiplicative weights.
In the sequel, $v$ and $v_{j}$ for $j\geq 0$, always stand for
submultiplicative weights if nothing else is stated.

We also remark that in the literature it is common to define submultiplicative
weights as (1.43) should hold, without the condition $v(-x)=v(x)$, i. e. that
$v$ does not have to be even (cf. e. g. [18, 20, 26, 29]). However, in the
sequel it is convenient for us to include this property in the definition.

There are several subclasses of $\mathscr{P}_{E}(\mathbf{R}^{d})$ which are
interesting for different reasons. Though our results later on are formulated
in background of $\mathscr{P}_{E}(\mathbf{R}^{d})$, we here mention some
subclasses which especially appear in time-frequency analysis. First we
observe the class $\mathscr{P}^{0}_{E}(\mathbf{R}^{d})$, which consists of all
$\omega\in\mathscr{P}_{E}(\mathbf{R}^{d})$ such that (1.46) holds for every
$r>0$.

The class $\mathscr{P}_{E}^{0}(\mathbf{R}^{d})$ is important when dealing with
spectral invariance for matrix or convolution operators on
$\ell^{2}(\mathbf{Z}^{d})$ (see e. g. [31]). If
$v\in\mathscr{P}_{E}(\mathbf{R}^{d})$ is submultiplicative, then
$v\in\mathscr{P}_{E}^{0}(\mathbf{R}^{d})$, if and only if

$\lim_{n\to\infty}v(nx)^{\frac{1}{n}}=1$ (1.47)

(see e. g. [24]). The condition (1.47) is equivalent to

$\lim_{n\to\infty}\frac{\log(v(nx))}{n}=0,$ (1.47)′

and is usually called the _GRS condition_ , or _Gelfand-Raikov-Shilov
condition_.

A more restrictive condition on $v$ compared to (1.47)′ is given by the
Beurling-Domar condition

$\sum_{n=1}^{\infty}\frac{\log(v(nx))}{n^{2}}<\infty.$ (1.48)

This condition is strongly linked to non quasi-analytic classes which contain
non-trivial compactly supported elements (see e. g. [30]). Any subexponential
submultiplicative weight satisfies the Beurling-Domar condition. That is,
suppose that $\theta\in(0,1)$ and that $v(x)=e^{r|x|^{\theta}}$,
$x\in\mathbf{R}^{d}$, then (1.48) is fulfilled. We let
$\mathscr{P}_{\operatorname{BD}}(\mathbf{R}^{d})$ be the set of all weights
which are moderated by submultiplicative weights which satisfy the Beurling-
Domar condition.

Finally we let $\mathscr{P}(\mathbf{R}^{d})$ be the set of all weights on
$\mathbf{R}^{d}$ which are moderated by polynomially bounded functions. That
is, $\omega\in\mathscr{P}(\mathbf{R}^{d})$, if and only if there are positive
constants $r$ and $C$ such that

$\omega(x+y)\leq C\omega(x)(1+|y|)^{r},\qquad x,y\in\mathbf{R}^{d}.$

Here we observe that $v(x)=(1+|x|)^{r}$ is submultiplicative.

Among these weight classes we have

$\mathscr{P}(\mathbf{R}^{d})\subsetneq\mathscr{P}_{\operatorname{BD}}(\mathbf{R}^{d})\subsetneq\mathscr{P}_{E}^{0}(\mathbf{R}^{d})\subsetneq\mathscr{P}_{E}(\mathbf{R}^{d}).$
(1.49)

In fact, it is clear that the ordering in (1.49) holds. On the other hand, if
$r>0$ and $\theta\in(0,1)$, then due to the submultiplicative weights

$\displaystyle e^{r|x|^{\theta}}$
$\displaystyle\in\mathscr{P}_{\operatorname{BD}}(\mathbf{R}^{d})\setminus\mathscr{P}(\mathbf{R}^{d}),$
(1.50) $\displaystyle e^{r|x|/\log(e+|x|)}$
$\displaystyle\in\mathscr{P}_{E}^{0}(\mathbf{R}^{d})\setminus\mathscr{P}_{\operatorname{BD}}(\mathbf{R}^{d}),$
$\displaystyle\text{and}\quad e^{r|x|}$
$\displaystyle\in\mathscr{P}_{E}(\mathbf{R}^{d})\setminus\mathscr{P}_{E}^{0}(\mathbf{R}^{d}),$

it also follows that the inclusions in (1.49) are strict.

We refer to [16, 29, 30, 50] for more facts about weights in time-frequency
analysis.

###  1.5. Mixed norm spaces of Lebesgue type

For every $p,q\in(0,\infty]$ and weight $\omega$ on $\mathbf{R}^{2d}$, we set

$\displaystyle\|F\|_{L_{(\omega)}^{p,q}(\mathbf{R}^{2d})}$
$\displaystyle\equiv\|G_{F,\omega,p}\|_{L^{q}(\mathbf{R}^{d})},$ where
$\displaystyle\quad G_{F,\omega,p}(\xi)$
$\displaystyle=\|F(\,\cdot\,,\xi)\omega(\,\cdot\,,\xi)\|_{L^{p}(\mathbf{R}^{d})}$
and $\displaystyle\|F\|_{L^{p,q}_{*,(\omega)}(\mathbf{R}^{2d})}$
$\displaystyle\equiv\|H_{F,\omega,q}\|_{L^{p}(\mathbf{R}^{d})},$ where
$\displaystyle\quad H_{F,\omega,q}(x)$
$\displaystyle=\|F(x,\,\cdot\,)\omega(x,\,\cdot\,)\|_{L^{q}(\mathbf{R}^{d})},$

when $F$ is (complex-valued) measurable function on $\mathbf{R}^{2d}$. Then
$L^{p,q}_{(\omega)}(\mathbf{R}^{2d})$
($L^{p,q}_{*,(\omega)}(\mathbf{R}^{2d})$) consists of all measurable functions
$F$ such that $\|F\|_{L_{(\omega)}^{p,q}}<\infty$
($\|F\|_{L^{p,q}_{*,(\omega)}}<\infty$).

In similar ways, let $\Omega_{1},\Omega_{2}$ be discrete sets, $\omega$ be a
positive function on $\Omega_{1}\times\Omega_{2}$ and
$\ell_{0}^{\prime}(\Omega_{1}\times\Omega_{2})$ be the set of all formal
(complex-valued) sequences $c=\\{c(j,k)\\}_{j\in\Omega_{1},k\in\Omega_{2}}$.
Then the discrete Lebesgue spaces, i.e. the Lebesgue sequence spaces

$\ell^{p,q}_{(\omega)}(\Omega_{1}\times\Omega_{2})\quad\text{and}\quad\ell^{p,q}_{*,(\omega)}(\Omega_{1}\times\Omega_{2})$

of mixed (quasi-)norm types consist of all
$c\in\ell_{0}^{\prime}(\Omega_{1}\times\Omega_{2})$ such that
$\|c\|_{\ell_{(\omega)}^{p,q}(\Omega_{1}\times\Omega_{2})}<\infty$
respectively
$\|c\|_{\ell_{*,(\omega)}^{p,q}(\Omega_{1}\times\Omega_{2})}<\infty$. Here

$\displaystyle\|c\|_{\ell_{(\omega)}^{p,q}(\Omega_{1}\times\Omega_{2})}$
$\displaystyle\equiv\|G_{F,\omega,p}\|_{\ell^{q}(\Omega_{2})},$ where
$\displaystyle\quad G_{c,p}(k)$
$\displaystyle=\|F(\,\cdot\,,k)\omega(\,\cdot\,,k)\|_{\ell^{p}(\Omega_{1})}$
and $\displaystyle\|c\|_{\ell^{p,q}_{*,(\omega)}(\Omega_{1}\times\Omega_{2})}$
$\displaystyle\equiv\|H_{c,\omega,q}\|_{\ell^{p}(\Omega_{1})},$ where
$\displaystyle\quad H_{c,q}(j)$
$\displaystyle=\|c(j,\,\cdot\,)\omega(j,\,\cdot\,)\|_{\ell^{q}(\Omega_{2})},$

when $c\in\ell_{0}^{\prime}(\Omega_{1}\times\Omega_{2})$.

###  1.6. Convolutions and multiplications for discrete Lebesgue spaces

Next we discuss extended Hölder and Young relations for multiplications and
convolutions on discrete Lebesgue spaces. The Hölder and Young conditions on
Lebesgue exponent are then

$\displaystyle\frac{1}{q_{0}}$
$\displaystyle\leq\frac{1}{q_{1}}+\frac{1}{q_{2}},$ (1.51) respectively
$\displaystyle\frac{1}{p_{0}}$
$\displaystyle\leq\frac{1}{p_{1}}+\frac{1}{p_{2}}-\max\left(1,\frac{1}{p_{1}},\frac{1}{p_{2}}\right).$
(1.52)

Notice that, when $p_{1},p_{2}\in(0,1)$, then (1.52) becomes
$p_{0}\geq\max\\{p_{1},p_{2}\\}$, while for $p_{1},p_{2}\geq 1$ it reduces to
the common Young condition

$1+\frac{1}{p_{0}}\leq\frac{1}{p_{1}}+\frac{1}{p_{2}}.$

The conditions on the weight functions are

$\displaystyle\omega_{0}(j)$ $\displaystyle\leq\omega_{1}(j)\omega_{2}(j),$
$\displaystyle\qquad j$ $\displaystyle\in\Lambda,$ (1.53) respectively
$\displaystyle\omega_{0}(j_{1}+j_{2})$
$\displaystyle\leq\omega_{1}(j_{1})\omega_{2}(j_{2}),$ $\displaystyle\qquad
j_{1},j_{2}$ $\displaystyle\in\Lambda,$ (1.54)

where $\Lambda$ is a lattice of the form

$\Lambda=\\{\,n_{1}e_{1}+\cdots+n_{d}e_{d}\,;\,(n_{1},\dots,n_{d})\in\mathbf{Z}^{d}\,\\},$

where $e_{1},\dots e_{d}$ is a basis for $\mathbf{R}^{d}$.

######  Proposition 1.5.

Let $p_{j},q_{j}\in(0,\infty]$, $j=0,1,2$, be such that (1.51) and (1.52)
hold, let $\Lambda\subseteq\mathbf{R}^{d}$ be a lattice and let $\omega_{j}$
be weights on $\Lambda$, $j=0,1,2$. Then the following is true:

  1. (1)

if (1.53) holds, then the map $(a_{1},a_{2})\mapsto a_{1}\cdot a_{2}$ from
$\ell_{0}(\Lambda)\times\ell_{0}(\Lambda)$ to $\ell_{0}(\Lambda)$ extends
uniquely to a continuous map from
$\ell^{q_{1}}_{(\omega_{1})}(\Lambda)\times\ell^{q_{2}}_{(\omega_{2})}(\Lambda)$
to $\ell^{q_{0}}_{(\omega_{0})}(\Lambda)$, and

$\|a_{1}\cdot
a_{2}\|_{\ell^{q_{0}}_{(\omega_{0})}}\leq\|a_{1}\|_{\ell^{q_{1}}_{(\omega_{1})}}\|a_{2}\|_{\ell^{q_{2}}_{(\omega_{2})}},\qquad
a_{j}\in\ell^{q_{j}}_{(\omega_{j})}(\Lambda),\ j=1,2\text{;}$ (1.55)

  2. (2)

if (1.54) holds, then the map $(a_{1},a_{2})\mapsto a_{1}*a_{2}$ from
$\ell_{0}(\Lambda)\times\ell_{0}(\Lambda)$ to $\ell_{0}(\Lambda)$ extends
uniquely to a continuous map from
$\ell^{p_{1}}_{(\omega_{1})}(\Lambda)\times\ell^{p_{2}}_{(\omega_{2})}(\Lambda)$
to $\ell^{p_{0}}_{(\omega_{0})}(\Lambda)$, and

$\|a_{1}*a_{2}\|_{\ell^{p_{0}}_{(\omega_{0})}}\leq\|a_{1}\|_{\ell^{p_{1}}_{(\omega_{1})}}\|a_{2}\|_{\ell^{p_{2}}_{(\omega_{2})}},\qquad
a_{j}\in\ell^{p_{j}}_{(\omega_{j})}(\Lambda),\ j=1,2\text{.}$ (1.56)

The assertion (1) in Proposition 1.5 is the standard Hölder’s inequality for
discrete Lebesgue spaces. The assertion (2) in that proposition is the usual
Young’s inequality for Lebesgue spaces on lattices in the case when
$p_{0},p_{1},p_{2}\in[1,\infty]$. A proof of a weighted version of Proposition
1.5 is given in Appendix A in [55].

##  2\. Modulation spaces, multiplications and convolutions

In this section we introduce modulation spaces, and recall their basic
properties, in particular in the context of Gelfand-Shilov spaces. Notice that
we permit the Lebesgue exponents to belong to the full interval $(0,\infty]$
instead of the most common choice $[1,\infty]$, and general moderate weights
which may have a (sub)exponential growth. Here we also recall some facts on
Gabor expansions for modulation spaces.

Then we deduce multiplication and convolution estimates on modulation spaces.
There are several approaches to multiplication and convolution in the case
when the involved Lebesgue exponents belong to $[1,\infty]$ (see [9, 17, 20,
33, 44, 49]). Here we consider the case when these exponents belong
$(0,\infty)$ (see also [1, 2, 26, 42, 43, 51]). In addition, and in order to
keep the survey style of our exposition, we focus on the bilinear case, and
refer to [55] for extension of these results to multi-linear products as well
as allowing the Lebesgue exponents to belong to the full interval
$(0,\infty]$.

###  2.1. Modulation spaces

The (classical) modulation spaces, essentially introduced in [17] by
Feichtinger are given in the following. (See e. g. [19] for definition of more
general modulation spaces.)

######  Definition 2.1.

Let $p,q\in(0,\infty]$, $\omega\in\mathscr{P}_{E}(\mathbf{R}^{2d})$ and
$\phi\in\Sigma_{1}(\mathbf{R}^{d})\setminus 0$.

  1. (1)

The _modulation space_ $M^{p,q}_{(\omega)}(\mathbf{R}^{d})$ consists of all
$f\in\Sigma_{1}^{\prime}(\mathbf{R}^{d})$ such that

$\|f\|_{M^{p,q}_{(\omega)}}\equiv\|V_{\phi}f\|_{L^{p,q}_{(\omega)}}$

is finite. The topology of $M^{p,q}_{(\omega)}(\mathbf{R}^{d})$ is defined by
the (quasi-)norm $\|\,\cdot\,\|_{M^{p,q}_{(\omega)}}$;

  2. (2)

The _modulation space (of Wiener amalgam type)_
$W^{p,q}_{(\omega)}(\mathbf{R}^{d})$ consists of all
$f\in\Sigma_{1}^{\prime}(\mathbf{R}^{d})$ such that

$\|f\|_{W^{p,q}_{(\omega)}}\equiv\|V_{\phi}f\|_{L^{p,q}_{*,(\omega)}}$

is finite. The topology of $W^{p,q}_{(\omega)}(\mathbf{R}^{d})$ is defined by
the (quasi-)norm $\|\,\cdot\,\|_{W^{p,q}_{(\omega)}}$.

######  Remark 2.2.

Modulation spaces possess several convenient properties. In fact, let
$p,q\in(0,\infty]$, $\omega\in\mathscr{P}_{E}(\mathbf{R}^{2d})$ and
$\phi\in\Sigma_{1}(\mathbf{R}^{d})\setminus 0$. Then the following is true
(see [17, 19, 20, 21, 26, 29] and their analyses for verifications):

  * •

the definitions of $M_{(\omega)}^{p,q}(\mathbf{R}^{d})$ and
$W_{(\omega)}^{p,q}(\mathbf{R}^{d})$ are independent of the choices of
$\phi\in\Sigma_{1}(\mathbf{R}^{d})\setminus 0$, and different choices give
rise to equivalent quasi-norms;

  * •

the spaces $M^{p,q}_{(\omega)}(\mathbf{R}^{d})$ and
$W^{p,q}_{(\omega)}(\mathbf{R}^{d})$ are quasi-Banach spaces which increase
with $p$ and $q$, and decrease with $\omega$. If in addition $p,q\geq 1$, then
they are Banach spaces;

  * •

If in addition $p,q\geq 1$, then the $L^{2}(\mathbf{R}^{d})$ scalar product,
$(\,\cdot\,,\,\cdot\,)_{L^{2}(\mathbf{R}^{d})}$, on
$\Sigma_{1}(\mathbf{R}^{d})\times\Sigma_{1}(\mathbf{R}^{d})$ is uniquely
extendable to dualities between $M^{p,q}_{(\omega)}(\mathbf{R}^{d})$ and
$M^{p^{\prime},q^{\prime}}_{(1/\omega)}(\mathbf{R}^{d})$, and between
$W^{p,q}_{(\omega)}(\mathbf{R}^{d})$ and
$W^{p^{\prime},q^{\prime}}_{(1/\omega)}(\mathbf{R}^{d})$. If in addition
$p,q<\infty$, then the dual spaces of $M^{p,q}_{(\omega)}(\mathbf{R}^{d})$ and
$W^{p,q}_{(\omega)}(\mathbf{R}^{d})$ can be identified with
$M^{p^{\prime},q^{\prime}}_{(1/\omega)}(\mathbf{R}^{d})$ respectively
$W^{p^{\prime},q^{\prime}}_{(1/\omega)}(\mathbf{R}^{d})$, through the form
$(\,\cdot\,,\,\cdot\,)_{L^{2}(\mathbf{R}^{d})}$;

  * •

if $\omega_{0}(x,\xi)=\omega(-\xi,x)$, then $\mathscr{F}$ on
$\Sigma_{1}^{\prime}(\mathbf{R}^{d})$ restricts to a homeomorphism from
$M^{p,q}_{(\omega)}(\mathbf{R}^{d})$ to
$W^{q,p}_{(\omega_{0})}(\mathbf{R}^{d})$.

  * •

The inclusions

$\displaystyle\Sigma_{1}(\mathbf{R}^{d})$ $\displaystyle\subseteq$
$\displaystyle
M^{p,q}_{(\omega)}(\mathbf{R}^{d}),W^{p,q}_{(\omega)}(\mathbf{R}^{d})$
$\displaystyle\subseteq$
$\displaystyle\Sigma_{1}^{\prime}(\mathbf{R}^{d})\quad$ when
$\displaystyle\quad\omega\in\mathscr{P}_{E}(\mathbf{R}^{2d}),$ (2.1)
$\displaystyle\mathcal{S}_{1}(\mathbf{R}^{d})$ $\displaystyle\subseteq$
$\displaystyle
M^{p,q}_{(\omega)}(\mathbf{R}^{d}),W^{p,q}_{(\omega)}(\mathbf{R}^{d})$
$\displaystyle\subseteq$
$\displaystyle\mathcal{S}_{1}^{\prime}(\mathbf{R}^{d})\quad$ when
$\displaystyle\quad\omega\in\mathscr{P}_{E}^{0}(\mathbf{R}^{2d})$ (2.2) and
$\displaystyle\mathscr{S}(\mathbf{R}^{d})$ $\displaystyle\subseteq$
$\displaystyle
M^{p,q}_{(\omega)}(\mathbf{R}^{d}),W^{p,q}_{(\omega)}(\mathbf{R}^{d})$
$\displaystyle\subseteq$
$\displaystyle\,\mathscr{S}^{\prime}(\mathbf{R}^{d})\quad$ when
$\displaystyle\quad\omega\in\mathscr{P}(\mathbf{R}^{2d}).$ (2.3)

are continuous. If in addition $p,q<\infty$, then these inclusions are dense.

We recall from [50] the embeddings (2.1)–(2.3), are essentially special cases
of certain characterizations of the Schwartz space, Gelfand-Shilov spaces and
their distribution spaces in terms of suitable unions and intersections of
modulation spaces. In fact, let $p,q\in(0,\infty]$ and $s\geq 1$ be fixed and
set

$v_{r,t}(x,\xi)=\begin{cases}e^{r(|x|^{\frac{1}{t}}+|\xi|^{\frac{1}{t}}))},&t\in\mathbf{R}_{+}\\\\[4.30554pt]
(1+|x|+|\xi|)^{r},&t=\infty.\end{cases}$ (2.4)

Then

$\displaystyle\Sigma_{s}(\mathbf{R}^{d})$
$\displaystyle=\bigcap_{r>0}M^{p,q}_{(v_{r,s})}(\mathbf{R}^{d})$
$\displaystyle=\bigcap_{r>0}W^{p,q}_{(v_{r,s})}(\mathbf{R}^{d}),$ (2.5)
$\displaystyle\mathcal{S}_{s}(\mathbf{R}^{d})$
$\displaystyle=\bigcup_{r>0}M^{p,q}_{(v_{r,s})}(\mathbf{R}^{d})$
$\displaystyle=\bigcup_{r>0}W^{p,q}_{(v_{r,s})}(\mathbf{R}^{d}),$ (2.6)
$\displaystyle\mathscr{S}(\mathbf{R}^{d})$
$\displaystyle=\bigcap_{r>0}M^{p,q}_{(v_{r,\infty})}(\mathbf{R}^{d})$
$\displaystyle=\bigcap_{r>0}W^{p,q}_{(v_{r,\infty})}(\mathbf{R}^{d}),$ (2.7)
$\displaystyle\mathscr{S}^{\prime}(\mathbf{R}^{d})$
$\displaystyle=\bigcup_{r>0}M^{p,q}_{(1/v_{r,\infty})}(\mathbf{R}^{d})$
$\displaystyle=\bigcup_{r>0}W^{p,q}_{(1/v_{r,\infty})}(\mathbf{R}^{d}),$ (2.8)
$\displaystyle\mathcal{S}_{s}^{\prime}(\mathbf{R}^{d})$
$\displaystyle=\bigcap_{r>0}M^{p,q}_{(1/v_{r,s})}(\mathbf{R}^{d})$
$\displaystyle=\bigcap_{r>0}W^{p,q}_{(1/v_{r,s})}(\mathbf{R}^{d})$ (2.9) and
$\displaystyle\Sigma_{s}^{\prime}(\mathbf{R}^{d})$
$\displaystyle=\bigcup_{r>0}M^{p,q}_{(1/v_{r,s})}(\mathbf{R}^{d})$
$\displaystyle=\bigcup_{r>0}W^{p,q}_{(1/v_{r,s})}(\mathbf{R}^{d}).$ (2.10)

The topologies of the spaces on the left-hand sides of (2.5)–(2.10) are
obtained by replacing each intersection by projective limit with respect to
$r>0$ and each union with inductive limit with respect to $r>0$.

The relations (2.5)–(2.10) are essentially special cases of [50, Theorem 3.9],
see also [32, 46, 47]. In order to be self-contained we here give a proof of
(2.6).

###### Proof of (2.6).

Since

$M^{\infty}_{(v_{2r,s})}(\mathbf{R}^{d})\subseteq
M^{p,q}_{(v_{r,s})}(\mathbf{R}^{d}),W^{p,q}_{(v_{r,s})}(\mathbf{R}^{d})\subseteq
M^{\infty}_{(v_{r,s})}(\mathbf{R}^{d}),$

it suffices to prove the result for $p=q=\infty$. Let
$\phi\in\Sigma_{1}(\mathbf{R}^{d})\setminus 0$ be fixed. First suppose that

$f\in
M^{\infty}_{(v_{r,s})}(\mathbf{R}^{d})=W^{\infty}_{(v_{r,s})}(\mathbf{R}^{d}).$

Then it follows from the definition of modulation space norm that (1.39) holds
for some $r>0$. By Remark 1.4 it follows that
$f\in\mathcal{S}_{s}(\mathbf{R}^{d})$, and we have proved

$\bigcup_{r>0}M^{\infty}_{(v_{r,s})}(\mathbf{R}^{d})\subseteq\mathcal{S}_{s}(\mathbf{R}^{d}).$
(2.11)

Suppose instead that $f\in\mathcal{S}_{s}(\mathbf{R}^{d})$. Then (1.39) holds
for some $r>0$, giving that $f\in M^{\infty}_{(v_{r,s})}(\mathbf{R}^{d})$.
Hence (2.11) holds with reversed inclusion, and the result follows. ∎

######  Example 2.3.

Let $p=q=1$ and $\omega=1$. Then
$M^{1,1}_{(\omega)}(\mathbf{R}^{d})=M^{1}(\mathbf{R}^{d})$ is the Feichtinger
algebra, probably the most prominent example of a modulation space. We refer
to a recent survey [35] for a detailed account on $M^{1}(\mathbf{R}^{d})$, and
to [14, Lemma 11] for a list of its basic properties.

Familiar examples arise when $p=q=2$. Then
$M_{(\omega)}^{2,2}(\mathbf{R}^{d})=M^{2}(\mathbf{R}^{d})=L^{2}(\mathbf{R}^{d})$,
and

$M_{(\omega_{s})}^{2,2}(\mathbf{R}^{d})=H^{s}(\mathbf{R}^{d}),\qquad
s\in\mathbf{R},$

where $\omega_{s}(\xi)=\langle\xi\rangle^{s}$, and $H^{s}(\mathbf{R}^{d})$ is
the Sobolev space (also known as the Bessel potential space) of distributions
$f\in\mathcal{S}^{\prime}(\mathbf{R}^{d})$ such that

$\|f\|_{H^{s}}^{2}:=\int_{\mathbf{R}^{d}}\langle\xi\rangle^{2s}|\widehat{f}(\xi)|^{2}d\xi<\infty,$

cf. [29, Proposition 11.3.1]. Furthermore, if
$v_{s}(x,\xi)=\langle(x,\xi)\rangle^{s},$ then
$M_{(v_{s})}^{2,2}(\mathbf{R}^{d})=Q_{s}(\mathbf{R}^{d})$, $s\in\mathbf{R},$
[7, Lemma 2.3]. Here $Q_{s}$ denotes the Shubin-Sobolev space, [45].

Finally we remark that modulation spaces can be conveniently discretized in
terms of Gabor expansions. In order for explaining some basic issues on this,
in similar ways as in Subsection 1.5 in [55], we limit ourself to the case
when the involved weights are moderated by subexponential functions. That is,
we suppose that $\omega$ in $M^{p,q}_{(\omega)}(\mathbf{R}^{d})$ satisfies

$\omega(x+y,\xi+\eta)\lesssim\omega(x,\xi)e^{r(|x|^{\frac{1}{s}}+|\xi|^{\frac{1}{s}})},$
(2.12)

for some $s>1$ and $r>0$. We observe that this implies that

$\Sigma_{s}(\mathbf{R}^{d})\subseteq
M^{p,q}_{(\omega)}(\mathbf{R}^{d})\subseteq\Sigma_{s}^{\prime}(\mathbf{R}^{d}),$
(2.13)

in vew of (1.42), (2.5) and (2.10). For more general approaches we refer to
[20, 28, 29, 43, 51].

Since $s>1$, it follows from Sections 1.3 and 1.4 in [34] that there are
$\phi,\psi\in\Sigma_{s}(\mathbf{R}^{d})$ with values in $[0,1]$ such that

$\displaystyle\operatorname{supp}\phi$
$\displaystyle\subseteq\Big{[}-\frac{3}{4},\frac{3}{4}\Big{]}^{d},$
$\displaystyle\quad\phi(x)$ $\displaystyle=1$ when $\displaystyle\quad x$
$\displaystyle\in\Big{[}-\frac{1}{4},\frac{1}{4}\Big{]}^{d}$ (2.14)
$\displaystyle\operatorname{supp}\psi$ $\displaystyle\subseteq[-1,1]^{d},$
$\displaystyle\quad\psi(x)$ $\displaystyle=1$ when $\displaystyle\quad x$
$\displaystyle\in\Big{[}-\frac{3}{4},\frac{3}{4}\Big{]}^{d}$ (2.15)

and

$\sum_{j\in\mathbf{Z}^{d}}\phi(\,\cdot\,-j)=1.$ (2.16)

Let $f\in\Sigma_{s}^{\prime}(\mathbf{R}^{d})$. Then $x\mapsto f(x)\phi(x-j)$
belongs to $\Sigma_{s}^{\prime}(\mathbf{R}^{d})$ and is supported in
$j+[-\frac{3}{4},\frac{3}{4}]^{d}$. Hence, by periodization it follows from
Fourier analysis that

$f(x)\phi(x-j)=\sum_{\iota\in\pi\mathbf{Z}^{d}}c(j,\iota)e^{i\langle
x,\iota\rangle},\qquad x\in j+[-1,1]^{d},$ (2.17)

where

$c(j,\iota)=2^{-d}(f,\phi(\,\cdot\,-j)e^{i\langle\,\cdot\,,\iota\rangle})=\left(\frac{\pi}{2}\right)^{\frac{d}{2}}V_{\phi}f(j,\iota),\qquad
j\in\mathbf{Z}^{d},\ \iota\in\pi\mathbf{Z}^{d}.$

Since $\psi=1$ on the support of $\phi$, (2.17) gives

$f(x)\phi(x-j)=\left(\frac{\pi}{2}\right)^{\frac{d}{2}}\sum_{\iota\in\pi\mathbf{Z}^{d}}V_{\phi}f(j,\iota)\psi(x-j)e^{i\langle
x,\iota\rangle},\qquad x\in\mathbf{R}^{d},$ (2.17)′

By (2.16) it now follows that

$\displaystyle f(x)$
$\displaystyle=\left(\frac{\pi}{2}\right)^{\frac{d}{2}}\sum_{(j,\iota)\in\Lambda}V_{\phi}f(j,\iota)\psi(x-j)e^{i\langle
x,\iota\rangle},\qquad x\in\mathbf{R}^{d},$ (2.18) where
$\displaystyle\Lambda$
$\displaystyle=\mathbf{Z}^{d}\times(\pi\mathbf{Z}^{d}),$ (2.19)

which is the _Gabor expansion_ of $f$ with respect to the _Gabor pair_
$(\phi,\psi)$ and lattice $\Lambda$, i. e. with respect to the _Gabor atom_
$\phi$ and the _dual Gabor atom_ $\psi$. Here the series converges in
$\Sigma_{s}^{\prime}(\mathbf{R}^{d})$. By duality and the fact that compactly
supported elements in $\Sigma_{s}(\mathbf{R}^{d})$ are dense in
$\Sigma_{s}^{\prime}(\mathbf{R}^{d})$ we also have

$f(x)=\left(\frac{\pi}{2}\right)^{\frac{d}{2}}\sum_{(j,\iota)\in\Lambda}V_{\psi}f(j,\iota)\phi(x-j)e^{i\langle
x,\iota\rangle},\qquad x\in\mathbf{R}^{d},$ (2.20)

with convergence in $\Sigma_{s}^{\prime}(\mathbf{R}^{d})$.

Let $T$ be a linear continuous operator from $\Sigma_{s}(\mathbf{R}^{d})$ to
$\Sigma_{s}^{\prime}(\mathbf{R}^{d})$ and let
$f\in\Sigma_{s}(\mathbf{R}^{d})$. Then it follows from (2.18) that

$(Tf)(x)=\left(\frac{\pi}{2}\right)^{\frac{d}{2}}\sum_{(j,\iota)\in\Lambda}V_{\phi}f(j,\iota)T(\psi(\,\cdot\,-j)e^{i\langle\,\cdot\,,\iota\rangle})(x)$

and

$T(\psi(\,\cdot\,-j)e^{i\langle\,\cdot\,,\iota\rangle})(x)=\left(\frac{\pi}{2}\right)^{\frac{d}{2}}\sum_{(k,\kappa)\in\Lambda}(V_{\phi}(T(\psi(\,\cdot\,-j)e^{i\langle\,\cdot\,,\iota\rangle})))(k,\kappa)\psi(x-k)e^{i\langle
x,\kappa\rangle}.$

A combination of these expansions show that

$(Tf)(x)=\left(\frac{\pi}{2}\right)^{\frac{d}{2}}\sum_{(j,\iota)\in\Lambda}(A\cdot
V_{\phi}f)(j,\iota)\psi(x-j)e^{i\langle x,\iota\rangle},$ (2.21)

where $A=(a(\boldsymbol{j},K))_{\boldsymbol{j},\boldsymbol{k}\in\Lambda}$ is
the $\Lambda\times\Lambda$-matrix, given by

$a(\boldsymbol{j},\boldsymbol{k})=\left(\frac{\pi}{2}\right)^{\frac{d}{2}}(T(\psi(\,\cdot\,-j)e^{i\langle\,\cdot\,,\iota\rangle}),\phi(\,\cdot\,-k)e^{i\langle\,\cdot\,,\kappa\rangle})_{L^{2}(\mathbf{R}^{d})}\\\\[4.30554pt]
\text{when}\quad\boldsymbol{j}=(j,\iota)\ \text{and}\
\boldsymbol{k}=(k,\kappa).$ (2.22)

By the Gabor analysis for modulation spaces we get the following restatement
of [55, Proposition 1.8]. We refer to [18, 20, 21, 22, 26, 28, 29, 51] for
details.

######  Proposition 2.4.

Let $s>1$, $p,q\in(0,\infty]$, $\omega\in\mathscr{P}_{E}(\mathbf{R}^{2d})$ be
such that (2.12) holds for some $r>0$,
$\phi,\psi\in\Sigma_{s}(\mathbf{R}^{d})$ with values in $[0,1]$ be such that
(2.14), (2.15) and (2.16) hold true, and let
$f\in\Sigma_{s}^{\prime}(\mathbf{R}^{d})$. Then the following is true:

  1. (1)

$f\in M^{p,q}_{(\omega)}(\mathbf{R}^{d})$, if and only if
$\|V_{\phi}f\|_{\ell_{(\omega)}^{p,q}(\mathbf{Z}^{d}\times\pi\mathbf{Z}^{d})}$;

  2. (2)

$f\in M^{p,q}_{(\omega)}(\mathbf{R}^{d})$, if and only if
$\|V_{\psi}f\|_{\ell_{(\omega)}^{p,q}(\mathbf{Z}^{d}\times\pi\mathbf{Z}^{d})}$;

  3. (3)

the quasi-norms

$f\mapsto\|V_{\phi}f\|_{\ell_{(\omega)}^{p,q}(\mathbf{Z}^{d}\times\pi\mathbf{Z}^{d})}\quad\text{and}\quad
f\mapsto\|V_{\psi}f\|_{\ell_{(\omega)}^{p,q}(\mathbf{Z}^{d}\times\pi\mathbf{Z}^{d})}$

are equivalent to $\|\,\cdot\,\|_{M^{p,q}_{(\omega)}}$.

The same holds true with $W^{p,q}_{(\omega)}$ and $\ell_{*,(\omega)}^{p,q}$ in
place of $M^{p,q}_{(\omega)}$ respectively $\ell_{(\omega)}^{p,q}$ at each
occurrence.

###  2.2. Multiplications and convolutions in modulation spaces

As a first step for approaching multiplications and convolutions for elements
in modulation spaces, we reformulate such products in terms of short-time
Fourier transforms. Let
$\phi_{0},\phi_{1},\phi_{2}\in\Sigma_{1}(\mathbf{R}^{d})$ be fixed such that

$\phi_{0}=(2\pi)^{-\frac{d}{2}}\phi_{1}\phi_{2}$ (2.23)

and let $f_{1},f_{2}\in\Sigma_{1}(\mathbf{R}^{d})$. Then the multiplication
$f_{1}f_{2}$ can be expressed by

$\displaystyle F_{0}(x,\xi)$
$\displaystyle=\big{(}F_{1}(x,\,\cdot\,)*F_{2}(x,\,\cdot\,)\big{)}(\xi).$
(2.24) where $\displaystyle F_{j}$ $\displaystyle=V_{\phi_{j}}f_{j},\qquad
j=0,1,2,$ (2.25)

In fact, by Fourier’s inversion formula we get

$\big{(}(V_{\phi_{1}}f_{1})(x,\,\cdot\,)*(V_{\phi_{2}}f_{2})(x,\,\cdot\,)\big{)}(\xi)\\\\[4.30554pt]
=(2\pi)^{-d}\iiint
f_{1}(y_{1})\overline{\phi_{1}(y_{1}-x)}f_{2}(y_{2})\overline{\phi_{2}(y_{2}-x)}e^{-{i\langle
y_{1},\xi-\eta\rangle}}e^{-i\langle
y_{2},\eta\rangle}\,dy_{1}dy_{2}d\eta\\\\[4.30554pt] =\int
f_{1}(y)\overline{\phi_{1}(y-x)}f_{2}(y)\overline{\phi_{2}(y-x)}e^{-i\langle
y,\xi\rangle}\,dy=(2\pi)^{\frac{d}{2}}(V_{\phi_{1}\phi_{2}}(f_{1}f_{2}))(x,\xi).$

We also observe that we may extract $f_{0}=f_{1}f_{2}$ by the formula

$f_{0}=(\|\phi_{0}\|_{L^{2}})^{-1}V_{\phi_{0}}^{*}F_{0},$ (2.26)

provided $\phi_{0}$ is not trivially equal to $0$.

In the same way, let $\phi_{0},\phi_{1},\phi_{2}\in\Sigma_{1}(\mathbf{R}^{d})$
be fixed such that

$\phi_{0}=(2\pi)^{\frac{d}{2}}\phi_{1}*\phi_{2},$ (2.27)

and let $f_{1},f_{2},g\in\Sigma_{1}(\mathbf{R}^{d})$. Then the convolution
$f_{1}*f_{2}$ can be expressed by

$\displaystyle F_{0}(x,\xi)$
$\displaystyle=\big{(}F_{1}(\,\cdot\,,\xi)*F_{2}(\,\cdot\,,\xi)\big{)}(x).$
(2.28)

where $F_{j}$ are given by (2.25), and that we may extract $f_{0}=f_{1}*f_{2}$
from (2.26).

Next we discuss convolutions and multiplications for modulation spaces, and
start with the following convolution result for modulation spaces. For
multiplications of elements in modulation spaces we need to swap the
conditions for the involved Lebesgue exponents compared to (1.51) and (1.52).
That is, these conditions become

$\displaystyle\frac{1}{p_{0}}$
$\displaystyle\leq\frac{1}{p_{1}}+\frac{1}{p_{2}},$
$\displaystyle\qquad\frac{1}{q_{0}}$
$\displaystyle\leq\frac{1}{q_{1}}+\frac{1}{q_{2}}-\max\left(1,\frac{1}{p_{0}},\frac{1}{q_{1}},\frac{1}{q_{2}}\right),$
(2.29) or $\displaystyle\frac{1}{p_{0}}$
$\displaystyle\leq\frac{1}{p_{1}}+\frac{1}{p_{2}},$
$\displaystyle\qquad\frac{1}{q_{0}}$
$\displaystyle\leq\frac{1}{q_{1}}+\frac{1}{q_{2}}-\max\left(1,\frac{1}{q_{1}},\frac{1}{q_{2}}\right).$
(2.30)

The conditions on the weight functions are

$\displaystyle\omega_{0}(x,\xi_{1}+\xi_{2})$
$\displaystyle\leq\omega_{1}(x,\xi_{1})\omega_{2}(x,\xi_{2}),$
$\displaystyle\qquad x,\xi_{1},\xi_{2}$ $\displaystyle\in\mathbf{R}^{d},$
(2.31) respectively $\displaystyle\omega_{0}(x_{1}+x_{2},\xi)$
$\displaystyle\leq\omega_{1}(x_{1},\xi)\omega_{2}(x_{2},\xi),$
$\displaystyle\qquad x_{1},x_{2},\xi$ $\displaystyle\in\mathbf{R}^{d}.$ (2.32)

######  Theorem 2.5.

Let $p_{j},q_{j}\in(0,\infty)$ and
$\omega_{j}\in\mathscr{P}_{E}(\mathbf{R}^{2d})$, $j=0,1,2$, be such that
(2.29) and (2.31) hold. Then $(f_{1},f_{2})\mapsto f_{1}f_{2}$ from
$\Sigma_{1}(\mathbf{R}^{d})\times\Sigma_{1}(\mathbf{R}^{d})$ to
$\Sigma_{1}(\mathbf{R}^{d})$ is uniquely extendable to a continuous map from
$M^{p_{1},q_{1}}_{(\omega_{1})}(\mathbf{R}^{d})\times
M^{p_{2},q_{2}}_{(\omega_{2})}(\mathbf{R}^{d})$ to
$M_{(\omega_{0})}^{p_{0},q_{0}}(\mathbf{R}^{d})$, and

$\|f_{1}f_{2}\|_{M_{(\omega_{0})}^{p_{0},q_{0}}}\lesssim\|f_{1}\|_{M_{(\omega_{1})}^{p_{1},q_{1}}}\|f_{2}\|_{M_{(\omega_{2})}^{p_{2},q_{2}}},\quad
f_{j}\in M_{(\omega_{j})}^{p_{j},q_{j}}(\mathbf{R}^{d}),\ j=1,2\text{.}$
(2.33)

######  Theorem 2.6.

Let $p_{j},q_{j}\in(0,\infty)$ and
$\omega_{j}\in\mathscr{P}_{E}(\mathbf{R}^{2d})$, $j=0,1,2$, be such that
(2.30) and (2.31) hold. Then $(f_{1},f_{2})\mapsto f_{1}f_{2}$ from
$\Sigma_{1}(\mathbf{R}^{d})\times\Sigma_{1}(\mathbf{R}^{d})$ to
$\Sigma_{1}(\mathbf{R}^{d})$ is uniquely extendable to a continuous map from
$W^{p_{1},q_{1}}_{(\omega_{1})}(\mathbf{R}^{d})\times
W_{(\omega_{2})}^{p_{2},q_{2}}(\mathbf{R}^{d})$ to
$W_{(\omega_{0})}^{p_{0},q_{0}}(\mathbf{R}^{d})$, and

$\|f_{1}f_{2}\|_{W_{(\omega_{0})}^{p_{0},q_{0}}}\lesssim\|f_{1}\|_{W_{(\omega_{1})}^{p_{1},q_{1}}}\|f_{2}\|_{W_{(\omega_{2})}^{p_{2},q_{2}}},\quad
f_{j}\in W_{(\omega_{j})}^{p_{j},q_{j}}(\mathbf{R}^{d}),\ j=1,2\text{.}$
(2.34)

The corresponding results for convolutions are the following. Here the
conditions on the involved Lebesgue exponents are swapped as

$\displaystyle\frac{1}{p_{0}}$
$\displaystyle\leq\frac{1}{p_{1}}+\frac{1}{p_{2}}-\max\left(1,\frac{1}{q_{0}},\frac{1}{p_{1}},\frac{1}{p_{2}}\right),$
$\displaystyle\qquad\frac{1}{q_{0}}$
$\displaystyle\leq\frac{1}{q_{1}}+\frac{1}{q_{2}}$ (2.35) or
$\displaystyle\frac{1}{p_{0}}$
$\displaystyle\leq\frac{1}{p_{1}}+\frac{1}{p_{2}}-\max\left(1,\frac{1}{p_{1}},\frac{1}{p_{2}}\right),$
$\displaystyle\qquad\frac{1}{q_{0}}$
$\displaystyle\leq\frac{1}{q_{1}}+\frac{1}{q_{2}}$ (2.36)

######  Theorem 2.7.

Let $p_{j},q_{j}\in(0,\infty)$ and
$\omega_{j}\in\mathscr{P}_{E}(\mathbf{R}^{2d})$, $j=0,1,2$, be such that
(2.32) and (2.36) hold. Then $(f_{1},f_{2})\mapsto f_{1}*f_{2}$ from
$\Sigma_{1}(\mathbf{R}^{d})\times\Sigma_{1}(\mathbf{R}^{d})$ to
$\Sigma_{1}(\mathbf{R}^{d})$ is uniquely extendable to a continuous map from
$M^{p_{1},q_{1}}_{(\omega_{1})}(\mathbf{R}^{d})\times
M^{p_{2},q_{2}}_{(\omega_{2})}(\mathbf{R}^{d})$ to
$M_{(\omega_{0})}^{p_{0},q_{0}}(\mathbf{R}^{d})$, and

$\|f_{1}*f_{2}\|_{M_{(\omega_{0})}^{p_{0},q_{0}}}\lesssim\|f_{1}\|_{M_{(\omega_{1})}^{p_{1},q_{1}}}\|f_{2}\|_{M_{(\omega_{2})}^{p_{2},q_{2}}},\quad
f_{j}\in M_{(\omega_{j})}^{p_{j},q_{j}}(\mathbf{R}^{d}),\ j=1,2\text{.}$
(2.37)

######  Theorem 2.8.

Let $p_{j},q_{j}\in(0,\infty)$ and
$\omega_{j}\in\mathscr{P}_{E}(\mathbf{R}^{2d})$, $j=0,1,2$, be such that
(2.32) and (2.35) hold. Then $(f_{1},f_{2})\mapsto f_{1}*f_{2}$ from
$\Sigma_{1}(\mathbf{R}^{d})\times\Sigma_{1}(\mathbf{R}^{d})$ to
$\Sigma_{1}(\mathbf{R}^{d})$ is uniquely extendable to a continuous map from
$W^{p_{1},q_{1}}_{(\omega_{1})}(\mathbf{R}^{d})\times
W^{p_{2},q_{2}}_{(\omega_{2})}(\mathbf{R}^{d})$ to
$W_{(\omega_{0})}^{p_{0},q_{0}}(\mathbf{R}^{d})$, and

$\|f_{1}*f_{2}\|_{W_{(\omega_{0})}^{p_{0},q_{0}}}\lesssim\|f_{1}\|_{W_{(\omega_{1})}^{p_{1},q_{1}}}\|f_{2}\|_{W_{(\omega_{2})}^{p_{2},q_{2}}},\quad
f_{j}\in W_{(\omega_{j})}^{p_{j},q_{j}}(\mathbf{R}^{d}),\ j=1,2\text{.}$
(2.38)

We observe that Theorems 3.2–3.5 in [55] are multi-linear versions of the
previous results. In particular, Theorems 2.5–2.8 (which are special cases of
Theorems 3.2–3.5 in [55]) are Fourier transformations of Theorems 2.7 and 2.8.
Hence it suffices to prove the last two theorems, cf. [55]. To shed some ideas
of the arguments, we give a proof in the unweighted case of Theorem 2.7. We
will use Proposition A.1 from Appendix A, which is a special case of [55,
Proposition 3.6].

###### Proof of Theorem 2.7.

Suppose $f_{j}\in\mathscr{S}(\mathbf{R}^{d})$,
$\phi_{j}\in\mathscr{S}(\mathbf{R}^{d})$, $j=0,1,2$ be such that

$f_{0}=f_{1}*f_{2}\quad\text{and}\quad\phi_{0}=(2\pi)^{\frac{d}{2}}\phi_{1}*\phi_{2}\neq
0,$

and let $F_{j}$ be the same as in (2.25). Then

$F_{0}(x,\xi)=(V_{\phi_{1}}f_{1}(\,\cdot\,,\xi)*V_{\phi_{2}}f_{2}(\,\cdot\,,\xi))(x),$

in view of (2.28).

We have

$0\leq\chi_{k_{1}+Q}*\chi_{k_{2}+Q}\leq\chi_{k_{1}+k_{2}+Q_{d,2}},\qquad
k_{1},k_{2}\in\mathbf{Z}^{d},$

where $Q_{d,r}$ is the cube

$Q_{d,r}=[0,r]^{d}\quad\text{and}\quad Q=Q_{d,1}=[0,1]^{d},$

and $\chi_{E}$ is the characteristic function with respect to the set $E$.

Set

$\displaystyle G(x,\xi)$
$\displaystyle=(|V_{\phi_{1}}f_{1}(\,\cdot\,,\xi)|*|V_{\phi_{2}}f_{2}(\,\cdot\,,\xi)|)(x),$
$\displaystyle a_{j}(k,\kappa)$
$\displaystyle=\|V_{\phi_{j}}f_{j}\|_{L^{\infty}((k,\kappa)+Q_{2d,1})},\quad
j=1,2,$ and $\displaystyle b(k,\kappa)$
$\displaystyle=\|G\|_{L^{\infty}((k,\kappa)+Q_{2d,1})}$

Then

$\displaystyle\|V_{\phi_{0}}^{*}F_{0}\|_{M^{p_{0},q_{0}}}$
$\displaystyle\asymp\|P_{\phi_{0}}F_{0}\|_{\mathsf{W}(\ell^{p_{0},q_{0}})}\lesssim\|F_{0}\|_{\mathsf{W}(\ell^{p_{0},q_{0}})}$
$\displaystyle\leq\|G\|_{\mathsf{W}(\ell^{p_{0},q_{0}})}\asymp\|b\|_{\ell^{p_{0},q_{0}}},$
(2.39) and $\displaystyle\|f_{j}\|_{M^{p_{j},q_{j}}}$
$\displaystyle\asymp\|a_{j}\|_{\ell^{p_{j},q_{j}}}$ (2.40)

in view of (A.5) and Proposition A.1 in Appendix A (see also [26, Theorem
3.3])).

By (2.28) we have

$\displaystyle G(x,\lambda)$
$\displaystyle\leq\sum_{k_{1},k_{2}\in\mathbf{Z}^{d}}a_{1}(k_{1},\lambda)a_{2}(k_{2},\lambda)(\chi_{k_{1}+Q}*\chi_{k_{2}+Q})(x)$
(2.41)
$\displaystyle\leq\sum_{k_{1},k_{2}\in\mathbf{Z}^{d}}a_{1}(k_{1},\lambda)a_{2}(k_{2},\lambda)\chi_{k_{1}+k_{2}+Q_{d,2}}(x).$

We observe that

$\chi_{k_{1}+k_{2}+Q_{d,2}}(x)=0\quad\text{when}\quad x\notin l+Q_{d},\
(k_{1},k_{2})\notin\Omega_{l},$

where

$\Omega_{l}=\\{\,(k_{1},k_{2})\in\mathbf{Z}^{2d}\,;\,l_{j}-2\leq
k_{1,j}+k_{2,j}\leq l_{j}+1\,\\},$

and

$k_{j}=(k_{j,1},\dots,k_{j,d})\in\mathbf{Z}^{d},\qquad
j=1,2,\quad\text{and}\quad l=(l_{1},\dots,l_{d})\in\mathbf{Z}^{d}.$

Hence, if $x=l$ in (2.41), we get

$b(l,\lambda)\leq\sum_{(k_{1},k_{2})\in\Omega_{l}}a_{1}(k_{1},\lambda)a_{2}(k_{2},\lambda)\\\\[4.30554pt]
\leq\sum_{m\in
I}(a_{1}(\,\cdot\,,\lambda)*a_{2}(\,\cdot\,,\lambda))(l-2e_{0}+m),$ (2.42)

where $e_{0}=(1,\dots,1)\in\mathbf{Z}^{d}$ and $I=\\{0,1,2,3\\}^{d}$.

If we apply the $\ell^{p_{0}}$ quasi-norm on (2.42) with respect to the $l$
variable, then Proposition 1.5 (2) and the fact that $I$ is finite set give

$\|b(\,\cdot\,,\lambda)\|_{\ell^{p_{0}}}\leq\left\|\sum_{m\in
I}(a_{1}(\,\cdot\,,\lambda)*a_{2}(\,\cdot\,,\lambda))(\,\cdot\,-2e_{0}+m)\right\|_{\ell^{p_{0}}}\\\\[4.30554pt]
\leq\sum_{m\in
I}\|(a_{1}(\,\cdot\,,\lambda)*a_{2}(\,\cdot\,,\lambda))(\,\cdot\,-2e_{0}+m)\|_{\ell^{p_{0}}}\\\\[4.30554pt]
\asymp\|a_{1}(\,\cdot\,,\lambda)*a_{2}(\,\cdot\,,\lambda)\|_{\ell^{p_{0}}}\\\\[4.30554pt]
\leq\|a_{1}(\,\cdot\,,\lambda)\|_{\ell^{p_{1}}}\|a_{2}(\,\cdot\,,\lambda)\|_{\ell^{p_{2}}}.$

By applying the $\ell^{q_{0}}$ quasi-norm and using Proposition 1.5 (1) we now
get

$\|b\|_{\ell^{p_{0},q_{0}}}\lesssim\|a_{1}\|_{\ell^{p_{1},q_{1}}}\|a_{2}\|_{\ell^{p_{2},q_{2}}}.$

This is the same as

$\|G\|_{L^{p_{0},q_{0}}}\lesssim\|F_{1}\|_{L^{p_{1},q_{1}}}\|F_{2}\|_{L^{p_{2},q_{2}}}.$

A combination of this estimate with (2.39) and (2.40) gives that $f_{1}*f_{2}$
is well-defined and that (2.37) holds.

The uniqueness now follows from that (2.37) holds for
$f_{1},f_{2}\in\mathscr{S}(\mathbf{R}^{d})$, and that
$\mathscr{S}(\mathbf{R}^{d})$ is dense in $M^{p,q}(\mathbf{R}^{d})$ when
$p,q<\infty$. ∎

##  3\. Gabor products and modulation spaces

In this section we give an illustration how the multiplication properties for
modulation spaces can be used when treating certain nonlinear problems. We
consider the Gabor product which is connected to such multiplication
properties. It is introduced in [14] in order to derive a phase space analogue
to the usual convolution identity for the Fourier transform (1.1). We will
prove a formula related to (2.24), and then use results from previous section
to extend the Gabor product initially defined on $M^{1}(\mathbf{R}^{2d})\times
M^{1}(\mathbf{R}^{2d})$ to some other spaces. Finally, we show how the Gabor
product gives rise to a phase-space formulation of the qubic Schrödinger
equation.

######  Definition 3.1.

Let $\phi\in M^{1}(\mathbf{R}^{d})\setminus 0$, and let $F_{1},F_{2}\in
M^{1}(\mathbf{R}^{2d})$. Then the Gabor product $\natural_{\phi}$ is given by

$\left(F_{1}\natural_{\phi}\,F_{2}\right)(x,\xi)\\\\[4.30554pt]
=(2\pi)^{-d}e^{-i\langle
x,\xi\rangle}\iiint_{\mathbf{R}^{3d}}\overline{\widehat{\phi}\left(\zeta-\xi\right)}e^{i\langle
x,\zeta\rangle}F_{1}(y,\eta)F_{2}(y,\zeta-\eta)\,dyd\eta d\zeta.$ (3.1)

In the proof of [14, Lemma 13] it is justified that the Gabor product in (3.1)
is well-defined, and that

$\natural_{\phi}:M^{1}(\mathbf{R}^{2d})\times M^{1}(\mathbf{R}^{2d})\to
M^{1}(\mathbf{R}^{2d})$

is a continuous map.

The Gabor product is particularly well-suited in the context of the STFT.

######  Theorem 3.2.

Let $\phi,\phi_{1},\phi_{2}\in
M^{1}(\mathbf{R}^{d})\backslash\left\\{0\right\\}$. Then

$(\phi_{2},\phi_{1})_{L^{2}(\mathbf{R}^{d})}V_{\phi}(f_{1}\cdot
f_{2})=(V_{\phi_{1}}f_{1})\natural_{\phi}(V_{\overline{\phi}_{2}}f_{2}),\quad
f_{1},f_{2}\in M^{1}(\mathbf{R}^{d}).$ (3.2)

Moreover, $V_{g}(f_{1}\cdot f_{2})\in M^{1}(\mathbf{R}^{2d})$.

###### Proof.

We have

$\displaystyle((V_{\phi_{1}}f_{1})$
$\displaystyle\natural_{\phi}(V_{\overline{\phi}_{2}}f_{2}))(x,\xi)$ (3.3)
$\displaystyle=(2\pi)^{-d}e^{-i\langle
x,\xi\rangle}\iint_{\mathbf{R}^{2d}}\overline{\widehat{\phi}(\zeta-\xi)}e^{i\langle
x,\zeta\rangle}G(y,\zeta)\,dyd\zeta,$ (3.4) where $\displaystyle G(y,\zeta)$
$\displaystyle=\int_{\mathbf{R}^{d}}(V_{\phi_{1}}f_{1})(y,\eta)(V_{\overline{\phi}_{2}}f_{2})(y,\zeta-\eta)\,d\eta.$

By Parseval’s formula we get

$\displaystyle G(y,\zeta)$
$\displaystyle=\int_{\mathbf{R}^{d}}(V_{\phi_{1}}f_{1})(y,\eta)(V_{\overline{\phi}_{2}}f_{2})(y,\zeta-\eta)\,d\eta$
$\displaystyle=\int_{\mathbf{R}^{d}}\mathscr{F}(f_{1}\overline{\phi_{1}(\,\cdot\,-y)})(\eta)\mathscr{F}(f_{2}{\phi_{2}(\,\cdot\,-y)})(\zeta-\eta)\,d\eta$
$\displaystyle=(\mathscr{F}(f_{1}\overline{\phi_{1}(\,\cdot\,-y)})\,,\,\mathscr{F}(\overline{f_{2}\,\phi_{2}(\,\cdot\,-y)}e^{i\langle\,\cdot\,,\zeta\rangle}))_{L^{2}(\mathbf{R}^{d})}$
$\displaystyle=(f_{1}\overline{\phi_{1}(\,\cdot\,-y)}\,,\,\overline{f_{2}\,\phi_{2}(\,\cdot\,-y)}e^{i\langle\,\cdot\,,\zeta\rangle})_{L^{2}(\mathbf{R}^{d})}$
$\displaystyle=\int_{\mathbf{R}^{d}}f_{1}(z)\overline{\phi_{1}(z-y)}f_{2}(z)\phi_{2}(z-y)e^{-i\langle
z,\zeta\rangle}\,dz.$

By inserting this into (3.4) and using Fubini’s theorem we get

$\displaystyle((V_{\phi_{1}}f_{1})$
$\displaystyle\natural_{\phi}(V_{\overline{\phi}_{2}}f_{2}))(x,\xi)$
$\displaystyle=(2\pi)^{-d}e^{-i\langle
x,\xi\rangle}\iint_{\mathbf{R}^{2d}}\overline{\widehat{\phi}(\zeta-\xi)}e^{-i\langle
z-x,\zeta\rangle}f_{1}(z)f_{2}(z)H(z)\,dzd\zeta,$ where $\displaystyle H(z)$
$\displaystyle=\int_{\mathbf{R}^{d}}\phi_{2}(z-y)\overline{\phi_{1}(z-y)}\,dy=(\phi_{2},\phi_{1})_{L^{2}}.$

Hence, by evaluating the integral with respect to $\zeta$, and using Fourier’s
inversion formula, we get

$\displaystyle((V_{\phi_{1}}f_{1})$
$\displaystyle\natural_{\phi}((V_{\overline{\phi}_{2}}f_{2})))(x,\xi)$
$\displaystyle=(2\pi)^{-\frac{d}{2}}e^{-i\langle
x,\xi\rangle}(\phi_{2},\phi_{1})_{L^{2}}\int_{\mathbf{R}^{d}}\overline{\phi(z-x)}e^{i\langle
x-z,\xi\rangle}f_{1}(z)f_{2}(z)\,dz$
$\displaystyle=(\phi_{2},\phi_{1})_{L^{2}}V_{\phi}(f_{1}f_{2})(x,\xi),$

which gives (3.2), and the result follows. ∎

The formula (3.2) is closely related to (2.24). In fact, the windows
$\phi_{j}\in\Sigma_{1}(\mathbf{R}^{d})$, $j=0,1,2$, in (2.24) should satisfy
the condition (2.23), while (3.2) is valid for arbitrary non-zero elements
from $M^{1}(\mathbf{R}^{d})$. For example, when $\phi=\phi_{1}=\phi_{2}$ and
$\|\phi\|_{L^{2}(\mathbf{R}^{d})}=1$, then (3.2) reduces to

$V_{\phi}(f_{1}\cdot
f_{2})=(V_{\phi}f_{1})\natural_{\phi}(V_{\phi}f_{2}),\qquad f_{1},f_{2}\in
M^{1}(\mathbf{R}^{d}),$ (3.5)

while (2.24) does not allow such choice of windows.

One of the main goals of [14] are extensions of the Gabor product to some
function spaces $\mathcal{F}_{j}(\mathbf{R}^{2d})$, $j=0,1,2$, so that
$\natural_{\phi}$ maps $\mathcal{F}_{1}\times\mathcal{F}_{2}$ into
$\mathcal{F}_{0}$, with:

$\|F_{1}\natural_{\phi}F_{2}\|_{\mathcal{F}_{0}}\leq
C\|F_{1}\|_{\mathcal{F}_{1}}\|F_{2}\|_{\mathcal{F}_{2}}.$ (3.6)

This can be considered as a phase space form of the Young convolution
inequality.

Next we discuss continuity of the Gabor product on certain spaces involving
superpositions of certain short-time Fourier transforms. In the end we deduce
properties similar to [14, Theorem 29]. Instead of modulation spaces of the
form $M^{p,q}_{(\omega)}(\mathbf{R}^{d})$, $p,q\in[1,\infty)$,
$\omega\in\mathscr{P}_{E}(\mathbf{R}^{2d})$, here we consider modulation
spaces of Wiener amalgam types $W^{p,q}_{(\omega)}(\mathbf{R}^{d})$, and allow
the ”quasi-Banach” choice for Lebesgue parameters, i. e. $p$ and $q$ are
allowed to be smaller than one.

Thus, in what follows we assume that $p,q\in(0,\infty)$,
$\omega\in\mathscr{P}_{E}(\mathbf{R}^{2d})$ is $v$-moderate, and consider
$L^{p,q}_{*,(\omega)}(\mathbf{R}^{2d})$ spaces rather than
$L^{p,q}_{(\omega)}(\mathbf{R}^{2d})$ which are treated in [14].

We need some additional notation. Let $s>1$, $N\in\mathbf{N}$ be given, and
let

$\mathscr{G}=\left\\{\phi_{n}=\overline{\phi_{n}}\,;~{}n\in\mathbf{N}\right\\}\subseteq\Sigma_{s}(\mathbf{R}^{d}),$

be an orthonormal basis of $L^{2}(\mathbf{R}^{d})$. Then let
$\mathcal{V}_{\mathscr{G},\omega}^{(N),p,q}(\mathbf{R}^{2d})$ be the closure
of

${\mathcal{V}}_{\mathscr{G}}^{(N)}(\mathbf{R}^{2d})=\left\\{\sum_{n=1}^{N}V_{\phi_{n}}f_{n}~{};~{}f_{n}\in
W^{1,1}_{(v)}(\mathbf{R}^{d})\right\\}$ (3.7)

with respect to the $L^{p,q}_{*,(\omega)}(\mathbf{R}^{2d})$ norm. In
particular, if $N=1$, this reduces to the closure
$\overline{P_{\phi}}(L^{p,q}_{*,(\omega)}(\mathbf{R}^{2d}))$ of

$P_{\phi}(L^{p,q}_{*,(\omega)}(\mathbf{R}^{2d}))=V_{\phi}(W^{1,1}_{(v)}(\mathbf{R}^{d}))$

in the $L^{p,q}_{*,(\omega)}(\mathbf{R}^{2d})$ norm.

By [14, Theorem 26], it follows that for every
$F\in\mathcal{V}_{\mathscr{G},\omega}^{(N),p,q}(\mathbf{R}^{2d})$ there exist
$f_{n}\in W_{(\omega)}^{p,q}(\mathbf{R}^{d})$, $n=1,2,\dots,N$, and such that

$F=\sum_{n=1}^{N}V_{\phi_{n}}f_{n}~{}.$ (3.8)

######  Theorem 3.3.

Let $p_{j},q_{j}\in(0,\infty)$ and
$\omega_{j}\in\mathscr{P}_{E}(\mathbf{R}^{2d})$, $j=0,1,2$, be such that
(2.30) and (2.31) hold, and let $\phi\in\Sigma_{s}(\mathbf{R}^{d})$, $s>1$.
Then the Gabor product $\natural_{\phi}$ from
${\mathcal{V}}_{\mathscr{G}}^{(N)}\left(\mathbf{R}^{2d}\right)\times{\mathcal{V}}_{\mathscr{G}}^{(N)}\left(\mathbf{R}^{2d}\right)$
to $W_{(v)}^{1,1}(\mathbf{R}^{2d})$, extends uniquely to a continuous map from
$\mathcal{V}_{\mathscr{G},\omega_{1}}^{(N),p_{1},q_{1}}(\mathbf{R}^{2d})\times\mathcal{V}_{\mathscr{G},\omega_{2}}^{(N),p_{2},q_{2}}(\mathbf{R}^{2d})$
to $\overline{P_{\phi}}(L^{p_{0},q_{0}}_{*,(\omega_{0})}(\mathbf{R}^{2d}))$,
and

$\|F_{1}\natural_{\phi}F_{2}\|_{L_{*,(\omega_{0})}^{p_{0},q_{0}}}\lesssim\|F_{1}\|_{L_{*,(\omega_{1})}^{p_{1},q_{1}}}~{}\|F_{2}\|_{L_{*,(\omega_{2})}^{p_{2},q_{2}}}~{},$
(3.9)

for all
$F_{j}\in\mathcal{V}_{\mathscr{G},\omega_{j}}^{(N),p_{j},q_{j}}(\mathbf{R}^{2d})$,
$j=1,2$.

In particular, if $F_{j}=V_{\phi}f_{j}$, $j=1,2$, and $\|\phi\|=1$, then (3.9)
reduces to

$\|V_{\phi}f_{1}\natural_{\phi}V_{\phi}f_{2}\|_{L_{*,(\omega_{0})}^{p_{0},q_{0}}}=\|f_{1}f_{2}\|_{W_{(\omega_{0})}^{p_{0},q_{0}}}\lesssim\|f_{1}\|_{W_{(\omega_{1})}^{p_{1},q_{1}}}~{}\|f_{2}\|_{W_{(\omega_{2})}^{p_{2},q_{2}}}~{}.$
(3.10)

We omit the proof which is a slight modification of the proof of Theorem 29 in
[14].

We end the paper by formally demonstrating how the Gabor product arises in a
phase space version of the cubic Schrödinger equation. Consider the elliptic
nonlinear Schrödinger equation (NLSE) given by

$i\frac{\partial\psi}{\partial t}+\Delta\psi+\lambda|\psi|^{2}\psi=0,$ (3.11)

subject to the initial condition:

$\psi(x,0)=\varphi(x).$

Here $\lambda=\pm 1$ stands for an attracting $(\lambda=+1)$ or repulsive
$(\lambda=-1)$ power-law nonlinearity, and the Laplacian is given by

$\Delta=\sum_{j=1}^{d}\frac{\partial^{2}}{\partial x_{j}^{2}}.$

Thus we consider $\psi=\phi(x,t)$ with $x\in\mathbf{R}^{d},$ and $t$ in an
open interval $I\subseteq\mathbf{R}$.

Using the following intertwining relations

$V_{\phi}(x_{j}\psi)=-D_{\xi_{j}}V_{g}(\psi),\qquad
V_{\phi}(D_{x_{j}}\psi)=\left(\xi_{j}+D_{x_{j}}\right)V_{\phi}\psi,$

$j=1,\cdots,d$, and assuming that $\phi$ is a real-valued window, we obtain
upon application of the STFT $V_{\phi}$ to (3.11) that

$i\frac{\partial F}{\partial
t}-\sum_{j=1}^{d}\left(\xi_{j}+D_{x_{j}}\right)^{2}F+\lambda\widetilde{F}\natural_{\phi}F\natural_{\phi}F=0.$
(3.12)

Here, $D_{x_{j}}=-i\frac{\partial}{\partial x_{j}}$,

$F(x,\xi,t)=V_{\phi}(\psi(\,\cdot\,,t))(x,\xi)\\\\[4.30554pt]
=(2\pi)^{-\frac{d}{2}}\int_{\mathbf{R}^{d}}\psi(y,t)\overline{\phi(y-x)}e^{-i\langle
y,\xi\rangle}\,dy,\quad x,\xi\in\mathbf{R}^{d},\ t\in\mathbf{R},$

and $\widetilde{F}$ is given by

$\widetilde{F}(x,\xi)=\overline{F(x,-\xi)}.$ (3.13)

By considering (3.12) the phase-space formulation of the initial value problem
may be well-posed for more general initial distributions. This means that the
phase-space formulation ”contains” the solutions of the standard NLSE, but it
is richer, as it admits other solutions. We refer to [11, 12, 13], where
phase-space extensions are explored in several different contexts.

Let us conclude by noticing that (3.12) contains the triple product. Thus, its
qualitative analysis calls for a multilinear extension of Theorems 2.6 and
3.3. Then the conditions (2.30) and (2.31) become more involved, see [55].
Such analysis demands a more technical tools and arguments and goes beyond the
scope of this survey article.

##  Appendix A Some properties of Wiener amalgam spaces

There are convenient characterizations of modulation spaces in the framework
of Gabor analysis.

Let $\omega_{0}\in\mathscr{P}_{E}(\mathbf{R}^{d})$,
$\omega\in\mathscr{P}_{E}(\mathbf{R}^{2d})$, $p,q,r\in(0,\infty]$,
$Q_{d}=[0,1]^{d}$ be the unit cube, and set for measurable $f$ on
$\mathbf{R}^{d}$,

$\|f\|_{\mathsf{W}^{r}(\omega_{0},\ell^{p})}\equiv\|a_{0}\|_{\ell^{p}(\mathbf{Z}^{d})}$
(A.1)

when

$a_{0}(j)\equiv\|f\cdot\omega_{0}\|_{L^{r}(j+Q_{d})},\qquad
j\in\mathbf{Z}^{d},$

and measurable $F$ on $\mathbf{R}^{2d}$,

$\|F\|_{\mathsf{W}^{r}(\omega,\ell^{p,q})}\equiv\|a\|_{\ell^{p,q}(\mathbf{Z}^{2d})}\quad\text{and}\quad\|F\|_{\mathsf{W}(\omega,\ell^{p,q}_{*})}\equiv\|a\|_{\ell^{p,q}_{*}(\mathbf{Z}^{2d})}$
(A.2)

when

$a(j,\iota)\equiv\|F\cdot\omega\|_{L^{r}((j,\iota)+Q_{2d})},\qquad
j,\iota\in\mathbf{Z}^{d}.$

The Wiener amalgam space

$\mathsf{W}^{r}(\omega_{0},\ell^{p})=\mathsf{W}^{r}(\omega_{0},\ell^{p}(\mathbf{Z}^{d}))$

consists of all measurable $f\in L^{r}_{loc}(\mathbf{R}^{d})$ such that
$\|F\|_{\mathsf{W}^{r}(\omega_{0},\ell^{p})}$ is finite, and the Wiener
amalgam spaces

$\mathsf{W}^{r}(\omega,\ell^{p,q})=\mathsf{W}^{r}(\omega,\ell^{p,q}(\mathbf{Z}^{2d}))\quad\text{and}\quad\mathsf{W}^{r}(\omega,\ell^{p,q}_{*})=\mathsf{W}^{r}(\omega,\ell^{p,q}_{*}(\mathbf{Z}^{2d}))$

consist of all measurable $F\in L^{r}_{loc}(\mathbf{R}^{2d})$ such that
$\|F\|_{\mathsf{W}^{r}(\omega,\ell^{p,q})}$ respectively
$\|F\|_{\mathsf{W}^{r}(\omega,\ell^{p,q}_{*})}$ are finite. We observe that
$\mathsf{W}^{r}(\omega_{0},\ell^{p})$ is often denoted by
$W(L^{r},L^{p}_{(\omega)})$ or $W(L^{r},L^{p}_{(\omega)})$ in the literature
(see e. e. [18, 20, 26, 42]).

The topologies are defined through their respectively quasi-norms in (A.1) and
(A.2). For conveniency we set

$\mathsf{W}(\omega,\ell^{p,q})=\mathsf{W}^{\infty}(\omega,\ell^{p,q})\quad\text{and}\quad\mathsf{W}(\omega,\ell^{p,q}_{*})=\mathsf{W}^{\infty}(\omega,\ell^{p,q}_{*}),$

and if in addition $\omega=1$, we set

$\mathsf{W}(\ell^{p,q})=\mathsf{W}(\omega,\ell^{p,q})\quad\text{and}\quad\mathsf{W}(\ell^{p,q}_{*})=\mathsf{W}(\omega,\ell^{p,q}_{*}).$

Obviously, $\mathsf{W}^{r}(\omega_{0},\ell^{p})$ and
$\mathsf{W}^{r}(\omega,\ell^{p,q})$ increase with $p,q$, decrease with $r$,
and

$\displaystyle\mathsf{W}(\omega,\ell^{p,q})$ $\displaystyle\hookrightarrow
L^{p,q}_{(\omega)}(\mathbf{R}^{2d})\cap\Sigma_{1}^{\prime}(\mathbf{R}^{2d})\hookrightarrow
L^{p,q}_{(\omega)}(\mathbf{R}^{2d})\hookrightarrow\mathsf{W}^{r}(\omega,\ell^{p,q})$
(A.3) and $\displaystyle\|\,\cdot\,\|_{\mathsf{W}^{r}(\omega,\ell^{p,q})}$
$\displaystyle\leq\|\,\cdot\,\|_{L^{p,q}_{(\omega)}}\leq\|\,\cdot\,\|_{\mathsf{W}(\omega,\ell^{p,q})},\qquad
r\leq\min(1,p,q).$ (A.4)

On the other hand, for modulation spaces we have

$f\in M^{p,q}_{(\omega)}(\mathbf{R}^{d})\quad\Leftrightarrow\quad V_{\phi}f\in
L^{p,q}_{(\omega)}(\mathbf{R}^{2d})\quad\Leftrightarrow\quad
V_{\phi}f\in\mathsf{W}^{r}(\omega,\ell^{p,q})$ (A.5)

with

$\|f\|_{M^{p,q}_{(\omega)}}=\|V_{\phi}f\|_{L^{p,q}_{(\omega)}}\asymp\|V_{\phi}f\|_{\mathsf{W}^{r}(\omega,\ell^{p,q})}.$
(A.6)

The same holds true with $W^{p,q}_{(\omega)}$, $L^{p,q}_{*,(\omega)}$ and
$\mathsf{W}(\omega,\ell^{p,q}_{*})$ in place of $M^{p,q}_{(\omega)}$,
$L^{p,q}_{(\omega)}$ and $\mathsf{W}(\omega,\ell^{p,q})$, respectively, at
each occurrence. (For $r=\infty$ , see [29] when $p,q\in[1,\infty]$, [26, 51]
when $p,q\in(0,\infty]$, and for $r\in(0,\infty]$, see [54].)

We have now the following result on the projection operator $P_{\phi}$ in
(1.20) when acting on Wiener amalgam spaces.

######  Proposition A.1.

Let $p,q\in(0,\infty]$ and $\phi\in\mathscr{S}(\mathbf{R}^{d})\setminus 0$.
Then $P_{\phi}$ from $\mathscr{S}^{\prime}(\mathbf{R}^{2d})$ to
$\mathscr{S}^{\prime}(\mathbf{R}^{2d})$, and $V_{\phi}^{*}$ from
$\mathscr{S}^{\prime}(\mathbf{R}^{2d})$ to
$\mathscr{S}^{\prime}(\mathbf{R}^{d})$ restrict to continuous mappings

$\displaystyle P_{\phi}:\mathsf{W}(\ell^{p,q}(\mathbf{Z}^{2d}))$
$\displaystyle\to V_{\phi}(M^{p,q}(\mathbf{R}^{d})),$ (A.7) $\displaystyle
P_{\phi}:\mathsf{W}(\ell^{p,q}_{*}(\mathbf{Z}^{2d}))$ $\displaystyle\to
V_{\phi}(W^{p,q}(\mathbf{R}^{d})),$ (A.8) $\displaystyle
V_{\phi}^{*}:\mathsf{W}(\ell^{p,q}(\mathbf{Z}^{2d}))$ $\displaystyle\to
M^{p,q}(\mathbf{R}^{d})$ (A.9) and $\displaystyle
V_{\phi}^{*}:\mathsf{W}(\ell^{p,q}_{*}(\mathbf{Z}^{2d}))$ $\displaystyle\to
W^{p,q}(\mathbf{R}^{d}).$ (A.10)

We refer to [55, Proposition 3.6] for the proof of Proposition A.1 and to [20,
22, 29, 42, 43, 55] for some facts about the operators $P_{\phi}$ and
$V_{\phi}^{*}$,

For $p,q\geq 1$, i. .e. the case when all spaces are Banach spaces, proofs of
Proposition A.1 can be found in e. g. [29] as well as in abstract forms in
[20]. In the general case when $p,q>0$, we refer to [26, 43], since proofs of
Proposition A.1 are essentially given there.

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