# Trace mappings on quasi-Banach modulation spaces and applications to pseudo-
differential operators of amplitude type

Joachim Toft  Department of Mathematics, Linnæus University, Växjö, Sweden
<EMAIL_ADDRESS>,  Divyang G. Bhimani  Department of Mathematics, Indian
Institute of Science Education and Research, Pune, India
<EMAIL_ADDRESS>and  Ramesh Manna  School of Mathematical
Sciences, National Institute of Science Education and Research Bhubaneswar,
Jatni, India<EMAIL_ADDRESS>
###### Abstract.

We deduce trace properties for modulation spaces (including certain Wiener-
amalgam spaces) of Gelfand-Shilov distributions. We use these results to show
that $\Psi$dos with amplitudes in suitable modulation spaces, agree with
normal type $\Psi$dos whose symbols belong to (other) modulation spaces. In
particular we extend and improve the style of trace results for modulation
spaces in [9, 45] to include quasi-Banach modulation spaces. We also apply our
results to obtain Schatten-von Neumann and nuclearity properties for $\Psi$dos
with amplidudes in modulation spaces, extending earlier work in [12, 41, 45,
51].

###### Key words and phrases:

Modulation spaces, Gelfand-Shilov spaces, Wiener amalgam spaces, Trace map,
amplitude, pseudo-differential operators

###### 2010 Mathematics Subject Classification:

35S05, 46F05, 42B35

##  0\. Introduction

In the paper we deduce continuity properties of trace mappings when acting on
extended classes of modulation spaces. Thereafter we apply these properties to
get continuity and identification properties for amplitude type pseudo-
differential operators with amplitudes in modulation spaces. In contrast to
most of earlier approaches, e. g. [21, 45] we allow the Lebesgue exponents of
the involved modulation spaces to stay in the full inverval $(0,\infty]$. In
particular, the modulation spaces in our investigations are quasi-Banach
spaces, which might not be Banach spaces. Note that these Lebesgue exponents
in [45] should belong to the smaller interval $[1,\infty]$, while in [21] they
should belong to $(0,\infty)$ and thereby not allowed to attain $\infty$. In
particular, our investigations include trace mapping properties of the
modulation space $M^{\infty,q}$, $q\in(0,1)$, while analogous investigations
in [21, 45] do not host these spaces.

A trace map is an operator which reduce the dimension of the domain for
functions or distributions, by fixing some coordinates. For example, let
$f(x_{1},x_{2})$ be a function which depends on $x_{1}\in\Omega_{1}$ and
$x_{2}\in\Omega_{2}$, let $z\in\Omega_{2}$ be fixed. Then the map
$\operatorname{Tr}_{z}$ which takes $(x_{1},x_{2})\mapsto f(x_{1},x_{2})$ into
$x_{1}\mapsto f(x_{1},z)$, i. e.,

$(\operatorname{Tr}_{z}f)(x_{1})=f(x_{1},z),$

can be considered as the archetype of trace mappings, provided $x_{1}\mapsto
f(x_{1},z)$ makes sense as a function. (See [34] and Section 1 for notations.)

Trace mappings appear in natural ways in different kinds of problems, e. g. in
boundary value problems of partial differential equations. If
$P(t,x,D_{t},D_{x})$ is a partial differential operator, and $f$ and $u_{0}$
are fixed functions or distributions, then

$\begin{cases}P(t,x,D_{t},D_{x})u(t,x)=f(x,t),&t>0,\\\\[4.30554pt]
u(0,x)=u_{0}(x),\end{cases}$

is an example of an initial value problem. It is expected that the solution
$u(t,x)$ should possess suitable continuity and differentiability properties
as well as the trace map which takes $u(t,x)$ into $u_{0}(x)=u(0,x)$ is well-
defined. See e. g. [40] and the references therein for more facts on this.

Another example where trace mappings appear naturally concerns pseudo-
differential operators of amplitude types. For any amplitude
$a\in\mathscr{S}(\mathbf{R}^{3d})$, the pseudo-differential operator
$\operatorname{Op}(a)$ is the linear and continuous map from
$\mathscr{S}(\mathbf{R}^{d})$ to $\mathscr{S}(\mathbf{R}^{d})$, given by

$\displaystyle(\operatorname{Op}(a)f)(x)$
$\displaystyle=(2\pi)^{-d}\iint_{\mathbf{R}^{2d}}a(x,y,\zeta)e^{i\langle
x-y,\zeta\rangle}f(y)\,dyd\zeta.$ Any such operator may in a unique way be
expressed as a pseudo-differential operator of standard or Kohn-Nirenberg type
with symbol in $\mathscr{S}(\mathbf{R}^{2d})$. That is, there is a unique
$a_{0}\in\mathscr{S}(\mathbf{R}^{2d})$ such that $\operatorname{Op}(a)$ above
is equal to $a_{0}(x,D)=\operatorname{Op}_{0}(a_{0})$, where
$\displaystyle(\operatorname{Op}_{0}(a_{0})f)(x)$
$\displaystyle=(2\pi)^{-d}\iint_{\mathbf{R}^{2d}}a_{0}(x,\zeta)e^{i\langle
x-y,\zeta\rangle}f(y)\,dyd\zeta.$

The drop of variable $y$ when passing from $a(x,y,\zeta)$ to $a_{0}(x,\zeta)$
implies that somewhere in the process, a trace map appear. Indeed, $a_{0}$ is
obtained by the formula

$a_{0}(x,\zeta)=(e^{i\langle
D_{\zeta},D_{y}\rangle}a)(x,y,\zeta)\Big{|}_{y=x}=(e^{i\langle
D_{\zeta},D_{y}\rangle}a)(x,x+y,\zeta)\Big{|}_{y=0}$

(see e. g. [34, Section 18.2]).

Trace mappings often possess convenient continuity properties when acting on
spaces of continuous functions. For example, the mappings

$\displaystyle\operatorname{Tr}_{z}$
$\displaystyle:C^{N}(\mathbf{R}^{d_{1}+d_{2}})$ $\displaystyle\to
C^{N}(\mathbf{R}^{d_{1}}),$ (0.1) $\displaystyle\operatorname{Tr}_{z}$
$\displaystyle:\mathscr{S}(\mathbf{R}^{d_{1}+d_{2}})$
$\displaystyle\to\mathscr{S}(\mathbf{R}^{d_{1}}),$ (0.2) and
$\displaystyle\operatorname{Tr}_{z}$
$\displaystyle:\Sigma_{1}(\mathbf{R}^{d_{1}+d_{2}})$
$\displaystyle\to\Sigma_{1}(\mathbf{R}^{d_{1}}),$ (0.3)

are continuous (and surjective). More sensitive situations appear when the
intended domains of $\operatorname{Tr}_{z}$ contain elements with lack of
continuity, or, more dreadful, host elements with heavy singularities.

A classical example on such situations concerns trace mappings when acting on
Sobolev spaces $H^{2}_{s}(\mathbf{R}^{d_{1}+d_{2}})$. For $s>\frac{d_{2}}{2}$
one has that the mappings (0.2) and (0.3) are uniquely extendable to a
continuous map

$\operatorname{Tr}_{z}:H^{2}_{s}(\mathbf{R}^{d_{1}+d_{2}})\to
H^{2}_{s_{0}}(\mathbf{R}^{d_{1}}),$ (0.4)

provided $s\geq s_{0}+\frac{d_{2}}{2}$ (see e. g. [2]).

The latter trace property was extended in [45] to modulation spaces which
include the Sobolev spaces above as special cases. We recall that the
modulation spaces $M^{p,q}_{(\omega)}(\mathbf{R}^{d})$ and
$W^{p,q}_{(\omega)}(\mathbf{R}^{d})$, introduced by Feichtinger in [14, 15]
and further developed in [17, 18, 19, 20, 26] are the sets of (Gelfand-Shilov)
distributions whose short-time Fourier transforms belong to the weighted and
mixed Lebesgue spaces $L^{p,q}_{(\omega)}(\mathbf{R}^{2d})$ respectively
$L^{p,q}_{*,(\omega)}(\mathbf{R}^{2d})$. Here $\omega$ is a weight function on
phase (or time-frequency shift) space and $p,q\in(0,\infty]$. Note that
$W^{p,q}_{(\omega)}(\mathbf{R}^{d})$ is also an example on Wiener-amalgam
spaces (cf. [17]).

Since the introduction, modulation spaces have entered several fields within
mathematics and science, e. g. the theory of pseudo-differential operators
(see e. g. [29, 41, 43, 48, 3, 9] and the references therein for more recent
progresses).

It follows from Theorem 3.2 in [45] that if $p,q\in[1,\infty]$ and

$\omega_{0}(x,\xi)\langle\eta\rangle^{t}\leq C\omega(x,y,\xi,\eta),\qquad
t\geq\frac{d_{2}}{q^{\prime}}$ (0.5)

for some constant $C>0$, with the latter inequality strict when $q>1$, then
(0.3) is uniquely extendable to a continuous map

$\displaystyle\operatorname{Tr}_{z}:M^{p,q}_{(\omega)}(\mathbf{R}^{d_{1}+d_{2}})$
$\displaystyle\to M^{p,q}_{(\omega_{0})}(\mathbf{R}^{d_{1}}),$ (0.6) that is,
$\displaystyle\|\operatorname{Tr}_{z}f\|_{M^{p,q}_{(\omega_{0})}}$
$\displaystyle\lesssim\|f\|_{M^{p,q}_{(\omega)}}.$ (0.7)

Here $\langle x\rangle\equiv(1+|x|^{2})^{\frac{1}{2}}$, $x\in\mathbf{R}^{d}$,
as usual. If

$p=q=2,\quad\omega(x,y,\xi,\eta)=\langle(\xi,\eta)\rangle^{s},\quad\omega_{0}(x,\xi)=\langle\xi\rangle^{s_{0}}\quad\text{and}\quad
s\geq s_{0}+\frac{d_{2}}{2},$

then (0.6) agrees with (0.4).

There are other extensions of (0.4). In [21, Theorem 3.3], Feichtinger, Huang
and Wang use uniform-frequency decomposition techniques to establish trace
properties on $\alpha$-modulation spaces, and thereby achieve trace properties
on modulation and Besov spaces as special cases (cf. [21, Theorem 3.1]). In
[39], Schneider deduce trace mapping results for Besov and Triebel-Lizorkin
spaces, allowing the involved Lebesgue exponents to belong to the full
interval $(0,\infty]$.

In Section 2 we extend the trace mapping result (0.6) in the sense of relaxing
the conditions of the involved weight functions, allowing the Lebesgue
exponents to belong to the full interval $(0,\infty]$, and complete (0.6) and
(0.7) with trace maps for $W^{p,q}_{(\omega)}$ spaces. Especially we prove

$\operatorname{Tr}_{z}:W^{p,q}_{(\omega)}(\mathbf{R}^{d_{1}+d_{2}})\to
W^{p,q}_{(\omega_{0})}(\mathbf{R}^{d_{1}})$

is continuous when $\omega$ and $\omega_{0}$ are the same as for (0.6). (See
Theorems 2.2 and 2.3.) The involved weight functions should satisfy conditions
of the form

$\displaystyle\omega(x+y)$ $\displaystyle\lesssim\omega(x)e^{r|y|}$ (0.8) for
some $r>0$, i. e., we permit general moderate weights (cf. [28]). For example,
for any $r\in\mathbf{R}$ and $\theta\in(0,1]$, we allow the weights
$(1+|x|)^{r}$ and $e^{r|x|^{\theta}}$. Note that in [21, 45] the condition
(0.8) is replaced by $\displaystyle\omega(x+y)$
$\displaystyle\lesssim\omega(x)\langle y\rangle^{r},$ (0.9)

for some $r\geq 0$, which is more restricted. In particular, weights like
$e^{r|x|^{\theta}}$ are not allowed in [21, 45].

The conditions on the weights have strong impact on the shape of modulation
spaces. For example, conditions of the form (0.9) imply that the modulation
space $M^{p,q}_{(\omega)}(\mathbf{R}^{d})$ stay between
$\mathscr{S}(\mathbf{R}^{d})$ and $\mathscr{S}^{\prime}(\mathbf{R}^{d})$,
while conditions of the form (0.8) imply that
$M^{p,q}_{(\omega)}(\mathbf{R}^{d})$ may contain the whole
$\mathscr{S}^{\prime}(\mathbf{R}^{d})$, or might be contained in
$\mathscr{S}(\mathbf{R}^{d})$. In this respect, in contrast to [21, 45], our
trace mapping results for modulation spaces in Section 2 also include ultra-
distributions which are outside $\mathscr{S}^{\prime}(\mathbf{R}^{d})$.

In particular we show that (0.6) holds for such weights and $p,q\in(0,\infty]$
after (0.5) is relaxed into

$\omega_{0}(x,\xi)\langle\eta\rangle^{t}\leq
C\omega(x,y,\xi,\eta)e^{r|y|},\qquad t\geq
d_{2}\left(\max\left(1,\frac{1}{p},\frac{1}{q}\right)-\frac{1}{q}\right),$
(0.5)′

for some $r\geq 0$, where the latter inequality in (0.5)′ should be strict
when $q>\min(p,1)$. (See Theorem 2.2.)

To reach such general results for modulation spaces, we first use Gabor
expansions to deduce the quasi-norm estimate (0.7) for elements in the
Gelfand-Shilov space $\Sigma_{1}(\mathbf{R}^{d})$ (which is dense in
$\mathscr{S}(\mathbf{R}^{d})$). Here convolution properties for weighted
discrete Lebesgue spaces (see (2.15) and (2.16) for details) play key roles to
achieved the desired estimates. Thereafter we deduce extensions of (0.6) to
the whole $M^{p,q}_{(\omega)}(\mathbf{R}^{d})$ in the Banach space case,
$p,q\geq 1$ by applying Hahn-Banach’s theorem. In similar way as in [45], some
critical cases when $p=\infty$ are obtained by using the _narrow convergence_
, a weaker form of convergence compared to norm convergence, but sufficiently
strong to guarantee needed uniqueness properties (see [41, 45]).

Extensions to the general case, $p,q\in(0,\infty]$ are then obtained by
applying suitable embedding results for modulation spaces, exploiting the fact
that $M^{p,q}_{(\omega)}(\mathbf{R}^{d})$ increases with $p$ and $q$. Finally
the uniqueness of (0.6) follows in the case $p,q<\infty$ by using the fact
that $\Sigma_{1}(\mathbf{R}^{d})$ is dense in
$M^{p,q}_{(\omega)}(\mathbf{R}^{d})$. For general $p$ and $q$, the uniqueness
of (0.6) is then reached by embedding $M^{p,q}_{(\omega)}(\mathbf{R}^{d})$
into other modulation spaces, where uniqueness assertions hold. More
precisely, we find suitable weights $\omega_{1}$ and $\omega_{0,1}$ such that
the diagram

$\begin{CD}M^{p,q}_{(\omega)}(\mathbf{R}^{d})@>{\operatorname{Tr}_{z}}>{}>M^{p,q}_{(\omega_{0})}(\mathbf{R}^{d_{1}})\\\
@V{{I}}V{}V@V{}V{I}V\\\
M^{1,1}_{(\omega_{1})}(\mathbf{R}^{d})@>{}>{\operatorname{Tr}_{z}}>M^{1,1}_{(\omega_{0,1})}(\mathbf{R}^{d_{1}})\end{CD}$
(0.10)

commutes. Here $I$ denotes continuous inclusions. The uniqueness of the first
map $\operatorname{Tr}_{z}$ in (0.10) then follows from the uniqueness of the
second one in (0.10). Note that these arguments remind on those in [39, Remark
3.1], where Schneider explain how trace mappings on Besov and Triebel-Lizorkin
spaces can be extended to allow the involved Lebesgue exponents to stay in the
full interval $(0,\infty]$.

In contrast to Theorems 3.1 and 3.3 in [21], our results do not include any
trace results for Besov or, more generally, those $\alpha$-modulation spaces
which are not modulation spaces. It is not obvious whether the methods in
Section 2 are well-designed for such investigations. On the other hand, the
restrictions that the Lebesgue exponents in Theorems 3.1 and 3.3 in [21] are
not allowed to attain $\infty$ is removed in Theorem 2.2 in Section 2. This
restriction might also be removed by using the ideas in [39, Remark 3.1] or
behind (0.10) in combination with embedding results in [24, 33, 53]. (See
Remark 2.9.)

In Section 3 we apply our trace mapping results in Section 2 to show that
pseudo-differential operators with amplitudes in suitable modulation spaces
can be formulated as pseudo-differential operators of Kohn-Nirenberg type with
symbols in other modulation spaces. For example, as a consequence of our
investigations it follows that if $p,q\in(0,\infty]$, $\omega$ and
$\omega_{0}$ are moderate weights such that

$\omega(x,x+z,\zeta+\eta,\xi-\eta,\eta,z)\asymp\omega_{0}(x,\zeta,\xi,z)\langle\eta\rangle^{t},\qquad
t\geq\frac{d}{q^{\prime}}$

with strict inequality when $q>1$, then

$\operatorname{Op}(M^{p,q}_{(\omega)}(\mathbf{R}^{3d}))=\operatorname{Op}_{0}(M^{p,q}_{(\omega_{0})}(\mathbf{R}^{2d})).$
(0.11)

(See Theorem 3.3.) In particular, for any $a\in
M^{p,q}_{(\omega)}(\mathbf{R}^{3d})$, there is a unique $a_{0}\in
M^{p,q}_{(\omega_{0})}(\mathbf{R}^{2d})$ such that
$\operatorname{Op}(a)=\operatorname{Op}_{0}(a_{0})$. On the other hand, for
any $a_{0}\in M^{p,q}_{(\omega_{0})}(\mathbf{R}^{2d})$, there exists an $a\in
M^{p,q}_{(\omega)}(\mathbf{R}^{3d})$ such that
$\operatorname{Op}(a)=\operatorname{Op}_{0}(a_{0})$. (See Proposition 3.4.)
Hence, the map which takes $a\in M^{p,q}_{(\omega)}(\mathbf{R}^{3d})$ into
$a_{0}\in M^{p,q}_{(\omega_{0})}(\mathbf{R}^{2d})$ (which is uniquely defined)
is surjective.

In order to deduce the surjectivity, a natural idea might be to choose
$a(x,y,\zeta)=a_{0}(x,\zeta)$ as candidate for $a$ above. On the other hand,
if $p<\infty$ and $a_{0}\in M^{p,q}_{(\omega_{0})}(\mathbf{R}^{2d})$, then
$(x,y,\zeta)\mapsto a_{0}(x,\zeta)$ fails to belong to
$M^{p,q}_{(\omega)}(\mathbf{R}^{3d})$. Hence the choice
$a(x,y,\zeta)=a_{0}(x,\zeta)$ does not work in this case. Instead the choice

$a(x,y,\zeta)=e^{-i\langle
D_{\zeta},D_{y}\rangle}(a_{0}(x,\zeta)\varphi(y-x))$

works when $\varphi$ is a suitable function such that $\varphi(0)=1$. (Cf.
Remark 2.10, and Theorem 3.3 and its proof.)

The identity (0.11) (supplied by Theorem 3.3) leads to that any continuity or
compactness property for
$\operatorname{Op}_{0}(M^{p,q}_{(\omega_{0})}(\mathbf{R}^{2d}))$ carry over to
the class $\operatorname{Op}(\mathcal{M}^{p,q}_{(\omega)}(\mathbf{R}^{3d}))$.
Here $\mathcal{M}^{p,q}_{(\omega)}(\mathbf{R}^{3d})$ is a modification of
$M^{p,q}(\mathbf{R}^{3d})$, obtained by suitable linear pullbacks of the
involved elements (see Section 3 for strict definition). In Theorem 0.1′ in
Section 3 we use [50, Theorem 3.1] to achieve such continuity and compactness
properties on modulation spaces, which are more general compared to existing
results in the literature (e. g. in [41, 45]). For example, the following
result is an immediate consequence of Theorem 0.1′ in Section 3, and is
obtained by choosing $p=\infty$ and $q\in(0,1]$ in that result. Here a common
condition for the involved weight function is

$\frac{\omega_{2}(x,\xi)}{\omega_{1}(z,\zeta)}\lesssim\omega(x,z,\zeta+\eta,\xi-\zeta-\eta,\eta,z-x),\qquad
x,z,\xi,\eta,\zeta\in\mathbf{R}^{d}.$ (0.12)

######  Theorem 0.1.

Let $p\in(0,\infty]$, $q\in(0,1]$, $\omega\in\mathscr{P}_{E}(\mathbf{R}^{6d})$
and $\omega_{1},\omega_{2}\in\mathscr{P}_{E}(\mathbf{R}^{2d})$ be such that
(0.12) holds. If $a\in\mathcal{M}^{\infty,q}_{(\omega)}(\mathbf{R}^{3d})$,
then $\operatorname{Op}(a)$ is continuous map from
$M^{p,q}_{(\omega_{1})}(\mathbf{R}^{d})$ to
$M^{p,q}_{(\omega_{2})}(\mathbf{R}^{d})$.

In the case $q=1$, $\omega=1$ and $\omega_{j}=1$, $j=1,2$, Theorem 0.1 is
proved already in [41]. For weights of polynomial type and Lebesgue exponents
are obeyed to stay in the restricted subinterval $[1,\infty]$ of $(0,\infty]$,
Theorem 0.1′ is essentially proved in [45].

Section 3 also includes some investigations on pseudo-differential operators
of amplitude types with symbols in modulation spaces of Wiener amalgam types.
For such operators we deduce continuity between suitable modulation spaces and
Wiener amalgam spaces (see Theorem 3.7).

In Section 3 we also combine Theorem 3.3 with suitable results in [50, 51] to
obtain detailed compactness results for pseudo-differential operators of
amplitude types. Especially the following Schatten-von Neumann and nuclearity
results are special cases of Theorem 0.2′ and Theorem 0.3′ in Section 3. (See
[50, Theorem 3.4] and [51, Theorem 4.2] for related results.)

######  Theorem 0.2.

Let $p,q\in(0,\infty]$ be such that $q\leq\min(p,p^{\prime})$,
$\omega\in\mathscr{P}_{E}(\mathbf{R}^{6d})$ and
$\omega_{1},\omega_{2}\in\mathscr{P}_{E}(\mathbf{R}^{2d})$ be such that (0.12)
holds. If $a\in\mathcal{M}^{p,q}_{(\omega)}(\mathbf{R}^{3d})$, then
$\operatorname{Op}(a)\in\mathscr{I}_{p}(M^{2}_{(\omega_{1})}(\mathbf{R}^{d}),M^{2}_{(\omega_{2})}(\mathbf{R}^{d}))$,
and

$\|\operatorname{Op}(a)\|_{\mathscr{I}_{p}(M^{2}_{(\omega_{1})},M^{2}_{(\omega_{2})})}\lesssim\|a\|_{\mathcal{M}^{p,q}_{(\omega)}},\quad
a\in\mathcal{M}^{p,q}_{(\omega)}(\mathbf{R}^{3d}).$

######  Theorem 0.3.

Let $p\in(0,1]$, $\omega\in\mathscr{P}_{E}(\mathbf{R}^{6d})$ and
$\omega_{1},\omega_{2}\in\mathscr{P}_{E}(\mathbf{R}^{2d})$ be such that (0.12)
holds. If $a\in\mathcal{M}^{p,p}_{(\omega)}(\mathbf{R}^{3d})$, then
$\operatorname{Op}(a)\in\mathscr{N}_{p}(M^{\infty}_{(\omega_{1})}(\mathbf{R}^{d}),M^{p}_{(\omega_{2})}(\mathbf{R}^{d}))$,
and

$\|\operatorname{Op}(a)\|_{\mathscr{N}_{p}(M^{\infty}_{(\omega_{1})},M^{p}_{(\omega_{2})})}\lesssim\|a\|_{\mathcal{M}^{p,p}_{(\omega)}},\quad
a\in\mathcal{M}^{p,p}_{(\omega)}(\mathbf{R}^{3d}).$

The paper is organizes as follows. In Section 1 we recall some basic facts for
Gelfand-Shilov, modulation spaces and pseudo-differential operators.
Thereafter we deduce trace results for modulation spaces in Section 2. Finally
we apply these trace results in Section 3 to transform common continuity and
compactness properties for pseudo-differential operators of standard or Kohn-
Nirenberg types into related properties for pseudo-differential operators of
amplitude types.

## Acknowledgement

The first author was supported by Vetenskapsrådet (Swedish Science Council),
within the project 2019-04890. The second author is thankful to
(DST/INSPIRE/04/2016/001507) and the third author is thankful to
(DST/INSPIRE/04/2019/001914) for research grants.

##  1\. Preliminaries

In this section we recall some facts on Gelfand-Shilov spaces, modulation
spaces and pseudo-differential operators. After introducing classes of weight
functions and mixed norm spaces of Lebesgue types, we recall some properties
of the Gelfand-Shilov space $\Sigma_{1}(\mathbf{R}^{d})$ and its distribution
space $\Sigma_{1}^{\prime}(\mathbf{R}^{d})$. Thereafter we consider a class of
modulation spaces which contains the classical modulation spaces, introduced
by Feichtinger in [15], but are not that general as in the more general
approach, given by Feichtinger in [17]. In the last part we recall the
definition of pseudo-differential operators and present some basic facts.

###  1.1. Weight functions

A _weight_ 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 $\omega$ on $\mathbf{R}^{d}$ is
called _moderate_ if there is a positive locally bounded function $v$ on
$\mathbf{R}^{d}$ such that

$\omega(x+y)\leq C\omega(x)v(y),\quad x,y\in\mathbf{R}^{d},$ (1.1)

for some constant $C\geq 1$. If $\omega$ and $v$ are weights on
$\mathbf{R}^{d}$ such that (1.1) holds, then $\omega$ is also called _$v$
-moderate_. The set of all moderate weights on $\mathbf{R}^{d}$ is denoted by
$\mathscr{P}_{E}(\mathbf{R}^{d})$.

We let $\mathscr{P}(\mathbf{R}^{d})$ be the set of all weights of polynomial
type. That is, (1.1) holds true for some (positive) polynomial $v$ on
$\mathbf{R}^{d}$. For $s\geq 1$ we also let
$\mathscr{P}_{E,s}(\mathbf{R}^{d})$ ($\mathscr{P}_{E,s}^{0}(\mathbf{R}^{d})$)
be the set of all weights on $\mathbf{R}^{d}$ such that for some $r>0$ (every
$r>0$) there is a constant $C>0$ such that

$\omega(x+y)\leq C\omega(x)e^{r|y|^{\frac{1}{s}}},\quad x,y\in\mathbf{R}^{d},$
(1.1)′

The weight $v$ on $\mathbf{R}^{d}$ is called _submultiplicative_ , if it is
even and (1.1) holds for $\omega=v$. From now on, $v$ always denotes a
submultiplicative weight if nothing else is stated. In particular, if (1.1)
holds and $v$ is submultiplicative, then it follows by straight-forward
computations that

$\begin{gathered}\frac{\omega(x)}{v(y)}\lesssim\omega(x+y)\lesssim\omega(x)v(y),\\\\[4.30554pt]
\quad v(x+y)\lesssim v(x)v(y)\quad\text{and}\quad v(x)=v(-x),\quad
x,y\in\mathbf{R}^{d}.\end{gathered}$ (1.2)

Here and in what follows we write $A(\theta)\lesssim B(\theta)$,
$\theta\in\Omega$, if there is a constant $c>0$ such that $A(\theta)\leq
cB(\theta)$ for all $\theta\in\Omega$. We also set $A(\theta)\asymp B(\theta)$
when $A(\theta)\lesssim B(\theta)\lesssim A(\theta)$.

If $\omega$ is a moderate weight on $\mathbf{R}^{d}$, then by [45, 46] and
above, there is a submultiplicative weight $v$ on $\mathbf{R}^{d}$ such that
(1.1) and (1.2) hold. Moreover if $v$ is submultiplicative on
$\mathbf{R}^{d}$, then

$1\lesssim v(x)\lesssim e^{r|x|}$ (1.3)

for some constant $r>0$ (cf. [28]). In particular, if $\omega$ is moderate,
then

$\omega(x+y)\lesssim\omega(x)e^{r|y|}\quad\text{and}\quad
e^{-r|x|}\leq\omega(x)\lesssim e^{r|x|},\quad x,y\in\mathbf{R}^{d}$ (1.4)

for some $r>0$.

###  1.2. Mixed quasi-normed spaces of Lebesgue types

Let $p,q\in(0,\infty]$, and let $\omega\in\mathscr{P}_{E}(\mathbf{R}^{2d})$.
Then $L^{p,q}_{(\omega)}(\mathbf{R}^{2d})$ and
$L^{p,q}_{*,(\omega)}(\mathbf{R}^{2d})$ consist of all measurable functions
$F$ on $\mathbf{R}^{2d}$ such that

$\displaystyle\|g_{1}\|_{L^{q}(\mathbf{R}^{d})}$ $\displaystyle<\infty,$ where
$\displaystyle\quad g_{1}(\xi)$
$\displaystyle\equiv\|F(\,\cdot\,,\xi)\omega(\,\cdot\,,\xi)\|_{L^{p}(\mathbf{R}^{d})}$
and $\displaystyle\|g_{2}\|_{L^{p}(\mathbf{R}^{d})}$ $\displaystyle<\infty,$
where $\displaystyle\quad g_{2}(x)$
$\displaystyle\equiv\|F(x,\,\cdot\,)\omega(x,\,\cdot\,)\|_{L^{q}(\mathbf{R}^{d})},$

respectively.

More generally, as in [50] we consider general classes of mixed quasi-normed
spaces of Lebesgue types, parameterized by

$p=(p_{1},\dots,p_{d})\in(0,\infty]^{d},\quad
q=(q_{1},\dots,q_{d})\in(0,\infty]^{d},$

$\sigma\in\operatorname{S}_{d}$ and
$\omega\in\mathscr{P}_{E}(\mathbf{R}^{d})$. Here $\operatorname{S}_{d}$ is the
set of permutations on $\\{1,\dots,d\\}$. In fact, let $p\in(0,\infty]^{d}$,
$\omega\in\mathscr{P}_{E}(\mathbf{R}^{d})$, and let
$\sigma\in\operatorname{S}_{d}$. Moreover, let $\Omega_{j}\subseteq\mathbf{R}$
be Borel-sets, $\mu_{j}$ be positive Borel measures on $\Omega_{j}$,
$j=1,\dots,d$, and let $\Omega=\Omega_{1}\times\cdots\times\Omega_{d}$ and
$\mu=\mu_{1}\otimes\cdots\otimes\mu_{d}$. For every measurable and complex-
valued function $f$ on $\Omega$, let $g_{j,\omega,\mu}$, $j=1,\dots,d-1$, be
defined inductively by

$\displaystyle g_{0,\omega,\mu}(x_{1},\dots,x_{d})$
$\displaystyle\equiv|f(x_{\sigma^{-1}(1)},\dots,x_{\sigma^{-1}(d)})\omega(x_{\sigma^{-1}(1)},\dots,x_{\sigma^{-1}(d)})|,$
$\displaystyle g_{k,\omega,\mu}(x_{k+1},\dots,x_{d})$
$\displaystyle\equiv\|g_{k-1,\omega,\mu}(\,\cdot\,,x_{k+1},\dots,x_{d})\|_{L^{p_{k}}(\mu_{k})},\quad
k=1,\dots,d-1,$ and let $\displaystyle\|f\|_{L^{p}_{\sigma,(\omega)}(\mu)}$
$\displaystyle\equiv\|g_{d-1,\omega,\mu}\|_{L^{p_{d}}(\mu_{d})}.$

The mixed quasi-norm space $L^{p}_{\sigma,(\omega)}(\mu)$ of Lebesgue type is
defined as the set of all $\mu$-measurable functions $f$ such that
$\|f\|_{L^{p}_{\sigma,(\omega)}(\mu)}<\infty$.

In the sequel we have $\Omega=\mathbf{R}^{d}$ and $d\mu=dx$, or
$\Omega=\Lambda$ and $\mu(j)=1$ when $j\in\Lambda$, where

$\displaystyle\Lambda$
$\displaystyle=\Lambda_{[\theta]}=T_{\theta}\mathbf{Z}^{d}\equiv\\{\,(\theta_{1}j_{1},\dots,\theta_{d}j_{d})\,;\,(j_{1},\dots,j_{d})\in\mathbf{Z}^{d}\,\\},$
(1.5) $\displaystyle\theta$
$\displaystyle=(\theta_{1},\dots,\theta_{d})\in\mathbf{R}_{*}^{d},\qquad\mathbf{R}_{*}=\mathbf{R}\setminus
0,$

and $T_{\theta}$ denotes the diagonal matrix with diagonal elements
$\theta_{1},\dots,\theta_{d}$. In the former case we set
$L^{p}_{\sigma,(\omega)}(\mu)=L^{p}_{\sigma,(\omega)}=L^{p}_{\sigma,(\omega)}(\mathbf{R}^{d})$,
and in the latter case we set
$L^{p}_{\sigma,(\omega)}(\mu)=\ell^{p}_{\sigma,(\omega)}(\Lambda)$.

For convenience we set

$L^{p,q}_{(\omega)}=L^{p,q}_{\sigma_{1},(\omega)}\qquad\text{and}\qquad
L^{p,q}_{*,(\omega)}=L^{q,p}_{\sigma_{2},(\omega)}$

when $p,q\in(0,\infty]^{d}$, $\omega\in\mathscr{P}_{E}(\mathbf{R}^{2d})$,
$\sigma_{1}\in\operatorname{S}_{2d}$ is the identity map and
$\sigma_{2}\in\operatorname{S}_{2d}$ is given by

$\sigma_{2}(j)=j+d\quad\text{and}\quad\sigma_{2}(j+d)=j,\qquad j=1,\dots,d.$
(1.6)

Later on it is common that $p$ above is split in some ways. If
$d_{j}\in\mathbf{N}$, $j=1,\dots,n$,

$p_{j}=(p_{j,1},\dots,p_{j,d_{j}})\in(0,\infty]^{d_{j}},\
j\in\\{1,\dots,n\\},$

and

$p=(p_{1},\dots,p_{n})=(p_{1,1},\dots,p_{1,d_{1}},\dots,p_{n,1},\dots,p_{n,d_{n}}),$

then set

$\displaystyle
L^{p_{1},\dots,p_{n}}_{(\omega)}(\mathbf{R}^{d_{1}+\cdots+d_{n}})$
$\displaystyle=L^{p}_{(\omega)}(\mathbf{R}^{d_{1}+\cdots+d_{n}})$ (1.7) and
$\displaystyle\|f\|_{L^{p_{1},\dots,p_{n}}_{(\omega)}}$
$\displaystyle=\|f\|_{L^{p_{1},\dots,p_{n}}_{(\omega)}(\mathbf{R}^{d_{1}+\cdots+d_{n}})}=\|f\|_{L^{p}_{(\omega)}},$
(1.8) when $f$ is complex-valued and measurable on
$\mathbf{R}^{d_{1}+\cdots+d_{n}}$. If in addition $d_{1}=\cdots=d_{n}=d$ for
some $d\geq 1$, then the space in (1.7) becomes $\displaystyle
L^{p}_{(\omega)}(\mathbf{R}^{nd})$
$\displaystyle=L^{p_{1},\dots,p_{n}}_{(\omega)}(\mathbf{R}^{nd})$ (1.7)′

with $p_{j}\in(0,\infty]^{d}$ for every $j=1,\dots,n$.

###  1.3. Gelfand-Shilov spaces

Let $0<h,s\in\mathbf{R}$ be fixed. Then $\mathcal{S}_{s;h}(\mathbf{R}^{d})$
consists of all $f\in C^{\infty}(\mathbf{R}^{d})$ such that

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

is finite. Here the supremum is taken over all $\alpha,\beta\in\mathbf{N}^{d}$
and $x\in\mathbf{R}^{d}$.

Obviously $\mathcal{S}_{s;h}\hookrightarrow\mathscr{S}$ is a Banach space
which increases with $h$ and $s$. Here and in what follows we use the notation
$A\hookrightarrow B$ when the topological spaces $A$ and $B$ satisfy
$A\subseteq B$ with continuous inclusion.

The _Gelfand-Shilov spaces_ $\mathcal{S}_{s}(\mathbf{R}^{d})$ and
$\Sigma_{s}(\mathbf{R}^{d})$ are the inductive and projective limits
respectively of $\mathcal{S}_{s;h}(\mathbf{R}^{d})$ with respect to $h>0$.
This implies that

$\mathcal{S}_{s}(\mathbf{R}^{d})=\bigcup_{h>0}\mathcal{S}_{s;h}(\mathbf{R}^{d})\quad\text{and}\quad\Sigma_{s}(\mathbf{R}^{d})=\bigcap_{h>0}\mathcal{S}_{s;h}(\mathbf{R}^{d}),$
(1.9)

and that the topology for $\mathcal{S}_{s}(\mathbf{R}^{d})$ is the strongest
possible one such that the inclusion map from
$\mathcal{S}_{s;h}(\mathbf{R}^{d})$ to $\mathcal{S}_{s}(\mathbf{R}^{d})$ is
continuous, for every choice of $h>0$. The space $\Sigma_{s}(\mathbf{R}^{d})$
is a Fréchet space with semi norms $\|\,\cdot\,\|_{\mathcal{S}_{s;h}}$, $h>0$.
Moreover, $\Sigma_{s}(\mathbf{R}^{d})\neq\\{0\\}$, if and only if $s>1/2$, and
$\mathcal{S}_{s}(\mathbf{R}^{d})\neq\\{0\\}$ if and only if $s\geq 1/2$ (cf.
[23, 36]).

Let $\mathcal{S}_{s;h}^{\prime}(\mathbf{R}^{d})$ be the dual of
$\mathcal{S}_{s;h}(\mathbf{R}^{d})$. Then the _Gelfand-Shilov distribution
spaces_ $\mathcal{S}_{s}^{\prime}(\mathbf{R}^{d})$ and
$\Sigma_{s}^{\prime}(\mathbf{R}^{d})$ are the projective and inductive limit
respectively of $\mathcal{S}_{s;h}^{\prime}(\mathbf{R}^{d})$ with respect to
$h>0$. This means that

$\mathcal{S}_{s}^{\prime}(\mathbf{R}^{d})=\bigcap_{h>0}\mathcal{S}_{s;h}^{\prime}(\mathbf{R}^{d})\quad\text{and}\quad\Sigma_{s}^{\prime}(\mathbf{R}^{d})=\bigcup_{h>0}\mathcal{S}_{s;h}^{\prime}(\mathbf{R}^{d}).$
(1.9)′

We remark that $\mathcal{S}_{s}^{\prime}(\mathbf{R}^{d})$ is the (strong) dual
of $\mathcal{S}_{s}(\mathbf{R}^{d})$ when $s\geq\frac{1}{2}$, and
$\Sigma_{s}^{\prime}(\mathbf{R}^{d})$ is the (strong) dual of
$\Sigma_{s}(\mathbf{R}^{d})$ when $s>\frac{1}{2}$ (cf. [36]). We also remark
that the form
$(\,\cdot\,,\,\cdot\,)_{L^{2}}=(\,\cdot\,,\,\cdot\,)_{L^{2}(\mathbf{R}^{d})}$
restricted to
$\mathcal{S}_{s}(\mathbf{R}^{d})\times\mathcal{S}_{s}(\mathbf{R}^{d})$
($\Sigma_{s}(\mathbf{R}^{d})\times\Sigma_{s}(\mathbf{R}^{d})$) is uniquely
extendable to a continuous map from
$\mathcal{S}_{s}^{\prime}(\mathbf{R}^{d})\times\mathcal{S}_{s}(\mathbf{R}^{d})$
($\Sigma_{s}^{\prime}(\mathbf{R}^{d})\times\Sigma_{s}(\mathbf{R}^{d})$) to
$\mathbf{C}$.

We have

$\mathcal{S}_{1/2}(\mathbf{R}^{d})\hookrightarrow\Sigma_{s_{1}}(\mathbf{R}^{d})\hookrightarrow\mathcal{S}_{s_{1}}(\mathbf{R}^{d})\hookrightarrow\Sigma_{s_{2}}(\mathbf{R}^{d})\hookrightarrow\mathscr{S}(\mathbf{R}^{d})\\\\[4.30554pt]
\hookrightarrow\mathscr{S}^{\prime}(\mathbf{R}^{d})\hookrightarrow\Sigma_{s_{2}}^{\prime}(\mathbf{R}^{d})\hookrightarrow\mathcal{S}_{s_{1}}^{\prime}(\mathbf{R}^{d})\hookrightarrow\Sigma_{s_{1}}^{\prime}(\mathbf{R}^{d})\hookrightarrow\mathcal{S}_{1/2}^{\prime}(\mathbf{R}^{d}),\qquad\frac{1}{2}<s_{1}<s_{2},$
(1.10)

with dense embeddings.

The Gelfand-Shilov spaces are invariant under several basic transformations.
For example they are invariant under translations, dilations and under
(partial) Fourier transformations. We also note that the map
$(f_{1},f_{2})\mapsto f_{1}\otimes f_{2}$ is continuous from
$\mathcal{S}_{s}(\mathbf{R}^{d_{1}})\times\mathcal{S}_{s}(\mathbf{R}^{d_{2}})$
to $\mathcal{S}_{s}(\mathbf{R}^{d_{1}+d_{2}})$, and similarly when each
$\mathcal{S}_{s}$ are replaced by $\Sigma_{s}$, $\mathcal{S}_{s}^{\prime}$ or
by $\Sigma_{s}^{\prime}$. (See also [52].)

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 map $\mathscr{F}$
extends uniquely to homeomorphisms on $\mathscr{S}^{\prime}(\mathbf{R}^{d})$,
$\mathcal{S}_{s}^{\prime}(\mathbf{R}^{d})$ and
$\Sigma_{s}^{\prime}(\mathbf{R}^{d})$, and restricts to homeomorphisms on
$\mathscr{S}(\mathbf{R}^{d})$, $\mathcal{S}_{s}(\mathbf{R}^{d})$ and
$\Sigma_{s}(\mathbf{R}^{d})$, and to a unitary operator on
$L^{2}(\mathbf{R}^{d})$.

There are several characterizations of Gelfand-Shilov spaces and their
distribution spaces (cf. [6, 13, 49] and the references therein). For example,
it follows from [6, 13] that $f\in\mathcal{S}_{s}(\mathbf{R}^{d})$
($f\in\Sigma_{s}(\mathbf{R}^{d})$), if and only if

$|f(x)|\lesssim
e^{-r|x|^{\frac{1}{s}}}\quad\text{and}\quad|\widehat{f}(\xi)|\lesssim
e^{-r|\xi|^{\frac{1}{s}}}$ (1.11)

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

Gelfand-Shilov spaces and their distribution spaces can also be characterized
by estimates on their short-time Fourier transforms. Let
$\phi\in\mathcal{S}_{s}(\mathbf{R}^{d})$ ($\phi\in\Sigma_{s}(\mathbf{R}^{d})$)
be fixed. Then the short-time Fourier transform of
$f\in\mathcal{S}_{s}^{\prime}(\mathbf{R}^{d})$ (of
$f\in\Sigma_{s}^{\prime}(\mathbf{R}^{d})$) with respect to $\phi$ is defined
by

$\displaystyle(V_{\phi}f)(x,\xi)$
$\displaystyle\equiv(2\pi)^{-\frac{d}{2}}(f,\phi(\,\cdot\,-x)e^{i\langle\,\cdot\,,\xi\rangle})_{L^{2}}.$
(1.12) We observe that $\displaystyle(V_{\phi}f)(x,\xi)$
$\displaystyle=\mathscr{F}(f\cdot\overline{\phi(\,\cdot\,-x)})(\xi)$ (1.12)′
(cf. [52]). If in addition $f\in L^{p}(\mathbf{R}^{d})$ for some
$p\in[1,\infty]$, then $\displaystyle(V_{\phi}f)(x,\xi)$
$\displaystyle=(2\pi)^{-\frac{d}{2}}\int_{\mathbf{R}^{d}}f(y)\overline{\phi(y-x)}e^{-i\langle
y,\xi\rangle}\,dy.$ (1.12)′′

In the next lemma we present characterizations of Gelfand-Shilov spaces and
their distribution spaces in terms of estimates on the short-time Fourier
transforms of the involved elements. The proof is omitted, since the first
part follows from [31], and the second part from [46, 49].

######  Lemma 1.1.

Let $p\in[1,\infty]$, $f\in\mathcal{S}^{\prime}_{1/2}(\mathbf{R}^{d})$,
$s\geq\frac{1}{2}$, $\phi\in\mathcal{S}_{s}(\mathbf{R}^{d})\setminus 0$
($\phi\in\Sigma_{s}(\mathbf{R}^{d})\setminus 0$) and

$v_{r}(x,\xi)=e^{r(|x|^{\frac{1}{s}}+|\xi|^{\frac{1}{s}})},\qquad r\geq 0.$

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

$\|V_{\phi}f\cdot v_{r}\|_{L^{p}}<\infty$ (1.13)

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

$\|V_{\phi}f/v_{r}\|_{L^{p}}<\infty$ (1.14)

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

######  Remark 1.2.

Evidently, if $d_{j}\in\mathbf{N}$ are such that $d=d_{1}+d_{2}+d_{3}\geq 1$,
$j=1,2,3$, $x_{j}\in\mathbf{R}^{d_{j}}$,
$x=(x_{1},x_{2},x_{3})\in\mathbf{R}^{d}$ and
$\xi=(\xi_{1},\xi_{2},\xi_{3})\in\mathbf{R}^{d}$, then we may replace $v_{r}$
in (1.13) and (1.14) by

$v_{r}(x,\xi)=e^{r(|x|^{\frac{1}{s}}+|\xi_{1}|^{\frac{1}{s}}+|\xi_{2}|^{\frac{1}{s}}+|\xi_{3}|^{\frac{1}{s}})},$

in order for Lemma 1.1 should hold true. In particular it follows that if
$\phi\in\Sigma_{1}(\mathbf{R}^{d})$ and
$f\in\mathcal{S}_{1/2}^{\prime}(\mathbf{R}^{d})$, then
$f\in\Sigma_{1}^{\prime}(\mathbf{R}^{d})$, if and only if

$(x,\xi)\mapsto V_{\phi}f(x,\xi)e^{-r(|x|+|\xi_{1}|+|\xi_{2}|+|\xi_{3}|)}$
(1.14)′

belongs to $L^{1}(\mathbf{R}^{d})$ for some $r>0$.

###  1.4. Modulation spaces

As in [50] we need a broader family of modulation spaces than what is
presented in [15, 35]. See also [17] for even more general modulation spaces.

######  Definition 1.3.

Let $p\in(0,\infty]^{2d}$, $\sigma\in\operatorname{S}_{2d}$,
$\omega\in\mathscr{P}_{E}(\mathbf{R}^{2d})$ and
$\phi\in\Sigma_{1}(\mathbf{R}^{d})\setminus 0$. Then the modulation space
$M^{p}_{\sigma,(\omega)}(\mathbf{R}^{d})$ consists of all
$f\in\Sigma_{1}^{\prime}(\mathbf{R}^{d})$ such that

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

is finite.

######  Remark 1.4.

Let $p,q\in(0,\infty]^{d}$, $\sigma_{1}\in\operatorname{S}_{2d}$ be the
identity map, $\sigma_{2}\in\operatorname{S}_{2d}$ be given by (1.6) and
$\omega\in\mathscr{P}_{E}(\mathbf{R}^{2d})$. Then set

$M^{p,q}_{(\omega)}\equiv M^{(p,q)}_{\sigma_{1},(\omega)}\quad\text{and}\quad
W^{p,q}_{(\omega)}\equiv M^{(q,p)}_{\sigma_{2},(\omega)}.$

We observe that $M^{p,q}_{(\omega)}(\mathbf{R}^{d})$ is a slight
generalization of the standard modulation spaces, introduced in [15], and
$W^{p,q}_{(\omega)}(\mathbf{R}^{d})$ is a Wiener amalgam space, considered in
[14].

In some situations later on one has that $p$ and $q$ are given by

$\displaystyle p$ $\displaystyle=$ $\displaystyle(p_{1},\dots,p_{n})$
$\displaystyle=(p_{1,1},\dots,p_{1,d_{1}},\dots,p_{n,1},\dots,p_{n,d_{n}}),$
$\displaystyle q$ $\displaystyle=$ $\displaystyle(q_{1},\dots,q_{n})$
$\displaystyle=(q_{1,1},\dots,q_{1,d_{1}},\dots,q_{n,1},\dots,q_{n,d_{n}}),$

for some $d_{j}\in\mathbf{N}$ $j=1,\dots,n$, where

$p_{j}=(p_{j,1},\dots,p_{j,d_{j}})\in(0,\infty]^{d_{j}},\quad
q_{j}=(q_{j,1},\dots,q_{j,d_{j}})\in(0,\infty]^{d_{j}}.$

It follows that

$M^{p,q}_{(\omega)}(\mathbf{R}^{d})=M^{p_{1},\dots,p_{n},q_{1},\dots,q_{n}}_{(\omega)}(\mathbf{R}^{d})=M^{p_{1},\dots,p_{n},q_{1},\dots,q_{n}}_{(\omega)}(\mathbf{R}^{d_{1}+\cdots+d_{n}})$

and

$W^{p,q}_{(\omega)}(\mathbf{R}^{d})=W^{p_{1},\dots,p_{n},q_{1},\dots,q_{n}}_{(\omega)}(\mathbf{R}^{d})=W^{p_{1},\dots,p_{n},q_{1},\dots,q_{n}}_{(\omega)}(\mathbf{R}^{d_{1}+\cdots+d_{n}})$

consist of all $f\in\Sigma_{1}^{\prime}(\mathbf{R}^{d})$ such that

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

are finite. Here $d=d_{1}+\dots+d_{n}$.

In our situations it is common that $n=3$ in Remark 1.4, i. e.,

$M^{p,q}_{(\omega)}(\mathbf{R}^{d_{1}+d_{2}+d_{3}})=M^{p_{1},p_{2},p_{3},q_{1},q_{2},q_{3}}_{(\omega)}(\mathbf{R}^{d_{1}+d_{2}+d_{3}}).$

It is also common that $d_{1}=d_{2}=d_{3}=d\geq 1$. In this situation we write

$M^{p,q}_{(\omega)}(\mathbf{R}^{d+d+d})=M^{p,q}_{(\omega)}(\mathbf{R}^{3d})=M^{p_{1},p_{2},p_{3},q_{1},q_{2},q_{3}}_{(\omega)}(\mathbf{R}^{3d})\phantom{.}$

and

$W^{p,q}_{(\omega)}(\mathbf{R}^{d+d+d})=W^{p,q}_{(\omega)}(\mathbf{R}^{3d})=W^{p_{1},p_{2},p_{3},q_{1},q_{2},q_{3}}_{(\omega)}(\mathbf{R}^{3d}).$

In what follows, the conjugate exponent $p^{\prime}\in(0,\infty]$ of
$p\in(0,\infty]$ is defined by

$p^{\prime}=\begin{cases}1&\text{when}\ p=\infty\\\\[4.30554pt]
\frac{p}{p-1}&\text{when}\ 1<p<\infty\\\\[4.30554pt] \infty&\text{when}\ p\leq
1.\end{cases}$

For any $p=(p_{1},\dots,p_{n})\in(0,\infty]^{n}$ and
$q=(q_{1},\dots,q_{n})\in(0,\infty]^{n}$ we set

$p^{-1}=\frac{1}{p}=(p_{1}^{-1},\dots,p_{n}^{-1}),\quad
p^{\prime}=(p_{1}^{\prime},\dots,p_{n}^{\prime}),\quad
p+q=(p_{1}+q_{1},\dots,p_{n}+q_{n})$

and

$p\leq q\quad(p<q)\qquad\text{when}\qquad p_{j}\leq q_{j}\quad(p_{j}<q_{j}),\
j=1,\dots,n.$

For $r\in(0,\infty]$ we also set

$p=r,\quad p\leq r\quad\text{respectively}\quad p<r$

when $p_{j}=r$, $p_{j}\leq r$ respectively $p_{j}<r$ for every $j=1,\dots,n$.

In the following proposition we list some basic properties of modulation
spaces. We omit the proof since the result follows by straight-forward
generalizations of the analysis in [22, 26, 47] (see also [18, 19, 20, 37,
38]).

######  Proposition 1.5.

Let $p,q\in(0,\infty]^{n}$, $r\in(0,1]$ be such that $r\leq\min(p,q)$,
$d_{j}\in\mathbf{N}$, $j=1,\dots,n$, $d=d_{1}+\cdots+d_{n}$,
$\omega,v\in\mathscr{P}_{E}(\mathbf{R}^{2d})$ be such that $\omega$ is
$v$-moderate, and let $\phi\in\Sigma_{1}(\mathbf{R}^{d})\setminus 0$.

  1. (1)

$\Sigma_{1}(\mathbf{R}^{d})\subseteq
M^{p,q}_{(\omega)}(\mathbf{R}^{d})\subseteq\Sigma_{1}^{\prime}(\mathbf{R}^{d})$.
If in addition $\max(p,q)<\infty$, then $\Sigma_{1}(\mathbf{R}^{d})$ is dense
in $M^{p,q}_{(\omega)}(\mathbf{R}^{d})$;

  2. (2)

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

  3. (3)

$M^{p,q}_{(\omega)}(\mathbf{R}^{d})$ is a quasi-Banach space which increase
with $p$ and $q$, and decrease with $\omega$. If in addition $\min(p,q)\geq
1$, then $M^{p,q}_{(\omega)}(\mathbf{R}^{d})$ is a Banach space;

  4. (4)

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 a duality between $M^{p,q}_{(\omega)}(\mathbf{R}^{d})$ and
$M^{p^{\prime},q^{\prime}}_{(1/\omega)}(\mathbf{R}^{d})$.

If in addition $p,q<(\infty,\dots,\infty)$, then the dual of
$M^{p,q}_{(\omega)}(\mathbf{R}^{d})$ can be identified with
$M^{p^{\prime},q^{\prime}}_{(1/\omega)}(\mathbf{R}^{d})$, through
$(\,\cdot\,,\,\cdot\,)_{L^{2}(\mathbf{R}^{d})}$;

  5. (5)

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

Similar facts hold true with $W^{p,q}_{(\omega)}$ spaces in place of
corresponding $M^{p,q}_{(\omega)}$ at each occurrence.

We set $M^{p,q}=M^{p,q}_{(\omega)}$ and $W^{p,q}=W^{p,q}_{(\omega)}$ when
$\omega=1$ and $p,q\in(0,\infty]^{n}$. We observe that

$M^{p_{0},q_{0}}=M^{p,q},\quad W^{p_{0},q_{0}}=W^{p,q}\quad\text{and}\quad
M^{p_{0}}=M^{p_{0},p_{0}}$

when

$p=(p_{0},\dots,p_{0})\in(0,\infty]^{n}\quad\text{and}\quad
q=(q_{0},\dots,q_{0})\in(0,\infty]^{n}.$

An important property for modulation spaces is that it is possible to
discretize them in terms of Gabor expansions. The following result is based on
[25, Theorem S] and the Gabor analysis in [22]. (See also Proposition 3.6 and
Theorem 3.7 in [47].)

######  Proposition 1.6.

Let $\omega,v\in\mathscr{P}_{E}(\mathbf{R}^{2d})$ be such that $\omega$ is
$v$-moderate, $p,q\in(0,\infty]^{d}$ and let $r\in(0,1]$ be such that
$r\leq\min(p,q)$. Then there is an $\varepsilon_{0}>0$ such that for every
$\varepsilon\in(0,\varepsilon_{0}]$, there are $\phi\in
M^{r}_{(v)}(\mathbf{R}^{d})$ and $\psi\in\Sigma_{1}(\mathbf{R}^{d})$ such that

$\displaystyle f$
$\displaystyle=\sum_{j,\iota\in\varepsilon\mathbf{Z}^{d}}(V_{\phi}f)(j,\iota)e^{i\langle\,\cdot\,,\iota\rangle}\psi(\,\cdot\,-j)$
$\displaystyle=\sum_{j,\iota\in\varepsilon\mathbf{Z}^{d}}(V_{\psi}f)(j,\iota)e^{i\langle\,\cdot\,,\iota\rangle}\phi(\,\cdot\,-j),\quad
f\in M^{\infty}_{(\omega)}(\mathbf{R}^{d}),$ (1.15)

with convergence of the series with respect to the weak∗ topology.
Furthermore,

$\|f\|_{M^{p,q}_{(\omega)}}\asymp\|V_{\phi}f\|_{\ell^{p,q}_{(\omega)}(\varepsilon\mathbf{Z}^{2d})}\asymp\|V_{\psi}f\|_{\ell^{p,q}_{(\omega)}(\varepsilon\mathbf{Z}^{2d})}.$

If in addition $\max(p,q)<\infty$, then the series in (1.15) converge
unconditionally to $f$ with respect to the $M^{p,q}_{(\omega)}$ norm.

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

###  1.5. Pseudo-differential operators

Next we discuss some issues in pseudo-differential calculus. Let
$\mathbf{M}(d,\Omega)$ be the set of all $d\times d$-matrices with entries in
the set $\Omega$, and let $a\in\Sigma_{1}(\mathbf{R}^{2d})$ and
$A\in\mathbf{M}(d,\mathbf{R})$ be fixed. Then the _pseudo-differential
operator_ $\operatorname{Op}_{A}(a)$ with _symbol_ $a$ is the linear and
continuous operator from $\Sigma_{1}(\mathbf{R}^{d})$ to
$\Sigma_{1}(\mathbf{R}^{d})$, defined by

$(\operatorname{Op}_{A}(a)f)(x)=(2\pi)^{-d}\iint_{\mathbf{R}^{2d}}a(x-A(x-y),\zeta)f(y)e^{i\langle
x-y,\zeta\rangle}\,dyd\zeta,$ (1.16)

when $f\in\Sigma_{1}(\mathbf{R}^{d})$. For general
$a\in\Sigma_{1}^{\prime}(\mathbf{R}^{2d})$, the pseudo-differential operator
$\operatorname{Op}_{A}(a)$ is defined as the linear and continuous operator
from $\Sigma_{1}(\mathbf{R}^{d})$ to $\Sigma_{1}^{\prime}(\mathbf{R}^{d})$
with distribution kernel given by

$K_{a,A}(x,y)=(2\pi)^{-\frac{d}{2}}(\mathscr{F}_{2}^{-1}a)(x-A(x-y),x-y).$
(1.17)

Here $\mathscr{F}_{2}F$ is the partial Fourier transform of
$F(x,y)\in\Sigma_{1}^{\prime}(\mathbf{R}^{2d})$ with respect to the $y$
variable. This definition makes sense, since the mappings

$\mathscr{F}_{2}\quad\text{and}\quad F(x,y)\mapsto F(x-A(x-y),x-y)$ (1.18)

are homeomorphisms on $\Sigma_{1}^{\prime}(\mathbf{R}^{2d})$. In particular,
the map $a\mapsto K_{a,A}$ is a homeomorphism on
$\Sigma_{1}^{\prime}(\mathbf{R}^{2d})$.

We set $\operatorname{Op}_{t}(a)=\operatorname{Op}_{t\cdot I}(a)$, when
$t\in\mathbf{R}$ and $I=I_{d}\in\mathbf{M}(d,\mathbf{R})$ is the $d\times d$
identity matrix. The normal or Kohn-Nirenberg representation, $a(x,D)$, is
obtained when $t=0$, and the Weyl quantization, $\operatorname{Op}^{w}(a)$, is
obtained when $t=\frac{1}{2}$. That is,

$a(x,D)=\operatorname{Op}_{0}(a)\quad\text{and}\quad\operatorname{Op}^{w}(a)=\operatorname{Op}_{1/2}(a).$

The following result explains the relationship between $a_{1}$ and $a_{2}$ in
the identity
$\operatorname{Op}_{A_{1}}(a_{1})=\operatorname{Op}_{A_{2}}(a_{2})$. We refer
to Propositions 1.1 and 2.8 [48] for the proof (see also [34] for background
ideas).

######  Proposition 1.7.

Let $p,q\in(0,\infty]$ $A,A_{1},A_{2}\in\mathbf{M}(d,\mathbf{R})$,
$\omega\in\mathscr{P}_{E}(\mathbf{R}^{4d})$ and let

$\omega_{A}(x,\xi,\eta,y)=\omega(x+Ay,\xi+A^{*}\eta,\eta,y).$ (1.19)

Then the following is true:

  1. (1)

$e^{i\langle AD_{\xi},D_{x}\rangle}$ is homeomorphic on
$\Sigma_{1}(\mathbf{R}^{2d})$ and uniquely extendable to a homeomorphism on
$\Sigma_{1}^{\prime}(\mathbf{R}^{2d})$;

  2. (2)

if $a_{1},a_{2}\in\Sigma_{1}^{\prime}(\mathbf{R}^{2d})$, then

$\operatorname{Op}_{A_{1}}(a_{1})=\operatorname{Op}_{A_{2}}(a_{2})\quad\Leftrightarrow\quad
e^{i\langle A_{1}D_{\xi},D_{x}\rangle}a_{1}(x,\xi)=e^{i\langle
A_{2}D_{\xi},D_{x}\rangle}a_{2}(x,\xi);$ (1.20)

  3. (3)

$e^{i\langle AD_{\xi},D_{x}\rangle}$ on $\Sigma_{1}^{\prime}(\mathbf{R}^{2d})$
restricts to a homoemorphism from $M^{p,q}_{(\omega)}(\mathbf{R}^{2d})$ to
$M^{p,q}_{(\omega_{A})}(\mathbf{R}^{2d})$.

Note here that the latter equality in (1.20) makes sense since it is
equivalent to

$e^{i\langle A_{2}x,\xi\rangle}\widehat{a}_{2}(\xi,x)=e^{i\langle
A_{1}x,\xi\rangle}\widehat{a}_{1}(\xi,x),$

and that the map $a\mapsto e^{i\langle Ax,\xi\rangle}a$ is continuous on
$\Sigma_{1}^{\prime}(\mathbf{R}^{2d})$ (cf. e. g. [54, 5]).

The next result is a slight extension of [50, Theorem 3.1], and follows from
[50, Theorem 3.1] and Proposition 1.7. The details are left for the reader.

######  Theorem 1.8.

Let $A\in\mathbf{M}(d,\mathbf{R})$, $\sigma\in\operatorname{S}_{2d}$,
$\omega_{1},\omega_{2}\in\mathscr{P}_{E}(\mathbf{R}^{2d})$ and
$\omega_{0}\in\mathscr{P}_{E}(\mathbf{R}^{4d})$ be such that

$\frac{\omega_{2}(x,\xi)}{\omega_{1}(y,\eta)}\lesssim\omega_{0}(x-A(x-y),\eta-A^{*}(\eta-\xi),\xi-\eta,y-x).$
(1.21)

Also let $p_{1},p_{2}\in(0,\infty]^{2d}$, $p,q\in(0,\infty]$ be such that

$\frac{1}{p_{2}}-\frac{1}{p_{1}}=\frac{1}{p}+\min\left(0,\frac{1}{q}-1\right),\quad
q\leq\min(p_{2})\leq\max(p_{2})\leq p,$ (1.22)

hold, and let $a\in M^{p,q}_{(\omega_{0})}(\mathbf{R}^{2d})$. Then
$\operatorname{Op}_{A}(a)$ from $\mathcal{S}_{1/2}(\mathbf{R}^{d})$ to
$\mathcal{S}_{1/2}^{\prime}(\mathbf{R}^{d})$ extends uniquely to a continuous
map from $M^{p_{1}}_{\sigma,(\omega_{1})}(\mathbf{R}^{d})$ to
$M^{p_{2}}_{\sigma,(\omega_{2})}(\mathbf{R}^{d})$, and

$\|\operatorname{Op}_{A}(a)f\|_{M^{p_{2}}_{\sigma,(\omega_{2})}}\lesssim\|a\|_{M^{p,q}_{(\omega_{0})}}\|f\|_{M^{p_{1}}_{\sigma,(\omega_{1})}},\quad
a\in M^{p,q}_{(\omega_{0})}(\mathbf{R}^{2d}),\ f\in
M^{p_{1}}_{\sigma,(\omega_{1})}(\mathbf{R}^{d}).$ (1.23)

By choosing $\sigma$ as the identity map, Theorem 1.8 gives continuity
properties for pseudo-differential operators acting on modulation spaces of
standard type, i. e. acting between spaces of the form
$M^{p,q}_{(\omega)}(\mathbf{R}^{d})$, where $p,q\in(0,\infty]$ or
$p,q\in(0,\infty]^{d}$ (see e. g. [41, 43, 45, 48] ). If instead $\sigma$
equals $\sigma_{2}$ in (1.6), Theorem 1.8 gives continuity properties for
pseudo-differential operators acting on Wiener amalgam type spaces of the form
$W^{p,q}_{(\omega)}(\mathbf{R}^{d})$. We remark that such continuity
properties were also established in [7].

Since $W^{p,q}_{(\omega)}$ spaces are Fourier images of $M^{p,q}_{(\omega)}$
spaces, any continuity result valid for $M^{p,q}_{(\omega)}$ spaces can be
transformed into a continuity result for $W^{p,q}_{(\omega)}$ spaces (see e.
g. [4] for ideas on such transitions).

For $A=0$, i. e. the standard or Kohn-Nirenberg case, (1.21) becomes

$\frac{\omega_{2}(x,\xi)}{\omega_{1}(y,\eta)}\lesssim\omega_{0}(x,\eta,\xi-\eta,y-x),$
(1.24)

and Theorem 1.8 implies

$\displaystyle\operatorname{Op}_{0}(a)\,:\,M^{q^{\prime},p^{\prime}}_{(\omega_{1})}(\mathbf{R}^{d})$
$\displaystyle\to$ $\displaystyle
M^{p,q}_{(\omega_{1})}(\mathbf{R}^{d}),\qquad a$ $\displaystyle\in
M^{p,q}_{(\omega_{0})}(\mathbf{R}^{2d}),$ $\displaystyle\ p,q$
$\displaystyle\in[1,\infty],\ q\leq p.$ (1.25) For slightly relaxed conditions
on $p$ and $q$ we also have
$\displaystyle\operatorname{Op}_{0}(a)\,:\,M^{q^{\prime},p^{\prime}}_{(\omega_{1})}(\mathbf{R}^{d})$
$\displaystyle\to$ $\displaystyle
W^{p,q}_{(\omega_{1})}(\mathbf{R}^{d}),\qquad a$ $\displaystyle\in
W^{p,q}_{(\omega_{0})}(\mathbf{R}^{2d}),$ $\displaystyle\ p,q$
$\displaystyle\in[1,\infty],$ (1.26)

concerning mapping properties for pseudo-differential operators with symbols
in Wiener amalgam type spaces $W^{p,q}_{(\omega_{0})}(\mathbf{R}^{2d})$. (See
e. g. [44, Theorem 3.9].) We observe that the condition $q\leq p$ in (1.25) is
removed in (1.26).

###  1.6. Pseudo-differential operators of amplitude types

Let $a\in\Sigma_{1}(\mathbf{R}^{3d})$. Then the pseudo-differential operator
$\operatorname{Op}(a)$ of _amplitude type_ with _amplitude_ $a$ is the linear
and continuous operator from $\Sigma_{1}(\mathbf{R}^{d})$ to
$\Sigma_{1}(\mathbf{R}^{d})$, defined by

$(\operatorname{Op}(a)f)(x)=(2\pi)^{-d}\iint_{\mathbf{R}^{2d}}a(x,y,\zeta)f(y)e^{i\langle
x-y,\zeta\rangle}\,dyd\zeta.$ (1.27)

The definition of $\operatorname{Op}(a)$ extends to more general $a$. For
example, in Section 3 we observe that $\operatorname{Op}(a)$ makes sense when
$a$ belongs to certain modulation spaces.

Let $A\in\mathbf{M}(d,\mathbf{R})$ be fixed. It is evident that the definition
of $\operatorname{Op}_{A}(a_{0})$ is a special case of $\operatorname{Op}(a)$,
since we may choose

$a(x,y,\zeta)=a_{0}(x-A(x-y),\zeta).$

On the other hand, it follows by Fourier inversion formula in combination with
kernel theorems for functions and distributions, it follows that any
continuous and linear operator from $\Sigma_{1}(\mathbf{R}^{d})$ to
$\Sigma_{1}^{\prime}(\mathbf{R}^{d})$ is given by
$\operatorname{Op}_{A}(a_{0})$ for a unique
$a_{0}\in\Sigma_{1}^{\prime}(\mathbf{R}^{2d})$. Consequently, the set of
linear and continuous operators from $\Sigma_{1}(\mathbf{R}^{d})$ to
$\Sigma_{1}^{\prime}(\mathbf{R}^{d})$ is not enlarged when passing from
operators of the form $\operatorname{Op}_{A}(a_{0})$ into the form
$\operatorname{Op}(a)$.

In particular, if $a\in\Sigma_{1}(\mathbf{R}^{3d})$, then there is a unique
$a_{0}(\mathbf{R}^{2d})\in\Sigma_{1}^{\prime}(\mathbf{R}^{2d})$ such that
$\operatorname{Op}(a)=\operatorname{Op}_{0}(a_{0})$. By straight-forward
computations it follows that

$\displaystyle\operatorname{Op}_{0}(a_{0})$
$\displaystyle=\operatorname{Op}(a)$ (1.28) when $\displaystyle
a_{0}(x,\zeta)$ $\displaystyle=(e^{i\langle
D_{\zeta},D_{y}\rangle}a)(x,y,\zeta)\Big{|}_{y=x}=(e^{i\langle
D_{\zeta},D_{y}\rangle}a)(x,x+y,\zeta)\Big{|}_{y=0}$ (1.29)

(see e. g. [34]).

##  2\. Trace properties of modulation spaces

In this section we deduce continuity properties of trace mappings on
modulation spaces. Especially we extend such mapping properties to modulation
spaces with general moderate weights.

More precisely, for any fixed $z\in\mathbf{R}^{d_{2}}$, we consider continuity
of the trace function map which takes a suitable function or distribution
$f(x_{1},x_{2},x_{3})$ into

$(\operatorname{Tr}_{z}f)(x_{1},x_{3})\equiv f(x_{1},z,x_{3}),\qquad
x_{1}\in\mathbf{R}^{d_{1}},\ x_{3}\in\mathbf{R}^{d_{3}}.$ (2.1)

Here $z\in\mathbf{R}^{d_{2}}$ is fixed and $x_{j}\in\mathbf{R}^{d_{j}}$ are
variables, $j=1,2,3$. By straight-forward computations it follows that
$\operatorname{Tr}_{z}$ is a linear and continuous map from
$C^{\infty}(\mathbf{R}^{d_{1}+d_{2}+d_{3}})$ to
$C^{\infty}(\mathbf{R}^{d_{1}+d_{3}})$, and that similar fact holds with
$\Sigma_{s}$, $\mathcal{S}_{s}$ or $\mathscr{S}$ in place of $C^{\infty}$ at
each occurrence.

######  Remark 2.1.

Let $d=d_{1}+d_{2}+d_{3}$, $z\in\mathbf{R}^{d_{2}}$ be fixed,
$x_{j}\in\mathbf{R}^{d_{j}}$ for $j=1,2,3$,

$f\in\Sigma_{1}(\mathbf{R}^{d})\quad\text{and}\quad
g_{z}(x_{0})=f(x_{1},z,x_{3}),\quad x_{0}=(x_{1},x_{3}).$

Also let $d_{0}=d_{1}+d_{3}$,

$\phi_{0}\in\Sigma_{1}(\mathbf{R}^{d_{0}})\setminus
0,\quad\phi_{2}\in\Sigma_{1}(\mathbf{R}^{d_{2}})\setminus
0\quad\text{and}\quad\phi(x_{1},x_{2},x_{3})=\phi_{0}(x_{0})\phi_{2}(x_{2}).$

If $\xi=(\xi_{1},\xi_{2},\xi_{3})\in\mathbf{R}^{d}$,
$\xi_{j}\in\mathbf{R}^{d_{j}}$ and $\xi_{0}=(\xi_{1},\xi_{3})$, then

$\displaystyle(V_{\phi_{0}}g_{z})(x_{0},\xi_{0})$
$\displaystyle=(2\pi)^{-\frac{d_{2}}{2}}\|\phi_{2}\|_{L^{2}}^{-2}\int_{\mathbf{R}^{2d_{2}}}V_{\phi_{0}}f_{Y}(x_{0},\xi_{0})\phi_{2}(z-y)e^{-i\langle
z,\eta\rangle}\,dY,$ (2.2) where $\displaystyle f_{Y}(x_{0})$
$\displaystyle=(V_{\phi_{2}}(f(x_{1},\,\cdot\,,x_{3})))(y,\eta),\qquad
Y=(y,\eta)\in\mathbf{R}^{2d_{2}}.$ (2.3)

In fact, (2.2) follows by first evaluating the integral with respect to $\eta$
and using Parseval’s formula, and thereafter integrate with respect to $y$.
For future references we notice that (2.2) is the same as

$(V_{\phi_{0}}g_{z})(x_{0},\xi_{0})\\\\[4.30554pt]
=(2\pi)^{-\frac{d_{2}}{2}}\|\phi_{2}\|_{L^{2}}^{-2}\iint_{\mathbf{R}^{2d_{2}}}V_{\phi}f(x_{1},y,x_{3},\xi_{1},\eta,\xi_{3})\phi_{2}(z-y)e^{-i\langle
z,\eta\rangle}\,dyd\eta.$ (2.4)

For modulation spaces we have the following trace result. Here the involved
Lebesgue exponents and weight functions should satisfy conditions of the form

$\displaystyle p_{0}=(p_{1},p_{3}),\quad p$
$\displaystyle=(p_{1},p_{2},p_{3}),\quad q_{0}=(q_{1},q_{3}),\quad
q=(q_{1},q_{2},q_{3}),$ (2.5)
$\displaystyle\max\left(\frac{1}{p_{0}},\frac{1}{q_{1}},\frac{1}{q_{2}},1\right)-\frac{1}{q_{2}}$
$\displaystyle\leq\frac{1}{r},$ (2.6)
$\sup_{\xi_{3}\in\mathbf{R}^{d_{3}}}\left(\left\|\sup_{(x,\xi_{1})\in\mathbf{R}^{d+d_{1}}}\left(\frac{\omega_{0}(x_{0},\xi_{0})e^{-r_{0}|x_{2}|}}{\omega(x,\xi_{1},\,\cdot\,,\xi_{3})}\right)\right\|_{L^{r}(\mathbf{R}^{d_{2}})}\right)<\infty$
(2.7)

and

$\omega(x,\xi)\lesssim\omega_{0}(x_{0},\xi_{0})e^{r_{0}(|x_{2}|+|\xi_{2}|)}.$
(2.8)

Here

$\displaystyle x$
$\displaystyle=(x_{1},x_{2},x_{3})\in\mathbf{R}^{d_{1}}\times\mathbf{R}^{d_{2}}\times\mathbf{R}^{d_{3}},$
$\displaystyle\quad x_{0}$ $\displaystyle=(x_{1},x_{3})$ (2.9)
$\displaystyle\xi$
$\displaystyle=(\xi_{1},\xi_{2},\xi_{3})\in\mathbf{R}^{d_{1}}\times\mathbf{R}^{d_{2}}\times\mathbf{R}^{d_{3}}$
$\displaystyle\quad\text{and}\quad\xi_{0}$ $\displaystyle=(\xi_{1},\xi_{3}).$

We observe that (2.7) is the same as

$\displaystyle\omega_{0}(x_{0},\xi_{0})e^{-r_{0}|x_{2}|}$
$\displaystyle\lesssim\omega(x,\xi)\vartheta(\xi_{2},\xi_{3}),\qquad
C_{\vartheta}\equiv\sup_{\xi_{3}\in\mathbf{R}^{d_{3}}}\|\vartheta(\,\cdot\,,\xi_{3})\|_{L^{r}}<\infty,$
(2.10) where $\vartheta\in\mathscr{P}_{E}(\mathbf{R}^{d_{2}+d_{3}})$ is given
by $\displaystyle\vartheta(\xi_{2},\xi_{3})$
$\displaystyle\equiv\sup_{x\in\mathbf{R}^{d}}\left(\sup_{\xi_{1}\in\mathbf{R}^{d_{1}}}\left(\frac{\omega_{0}(x_{0},\xi_{0})e^{-r_{0}|x_{2}|}}{\omega(x,\xi)}\right)\right),$
(2.11)

which indicates similarities between the conditions (2.7) and (2.8). We
observe

######  Theorem 2.2.

Let $d_{j}\geq 0$ be integers, $z\in\mathbf{R}^{d_{2}}$ be fixed,
$p_{j},q_{j}\in(0,\infty]^{d_{j}}$, $r\in(0,\infty]^{d_{2}}$, $j=1,2,3$, be
such that (2.6) holds and let
$\omega\in\mathscr{P}_{E}(\mathbf{R}^{2(d_{1}+d_{2}+d_{3})})$,
$\omega_{0}\in\mathscr{P}_{E}(\mathbf{R}^{2(d_{1}+d_{3})})$, $p$, $p_{0}$, $q$
and $q_{0}$ be such that (2.5) and (2.7) hold true for some $r_{0}\geq 0$.
Then the following is true:

  1. (1)

the map $\operatorname{Tr}_{z}$ from
$\Sigma_{1}(\mathbf{R}^{d_{1}+d_{2}+d_{3}})$ to
$\Sigma_{1}(\mathbf{R}^{d_{1}+d_{3}})$ extends uniquely to a continuous map
from $M^{p,q}_{(\omega)}(\mathbf{R}^{d_{1}+d_{2}+d_{3}})$ to
$M^{p_{0},q_{0}}_{(\omega_{0})}(\mathbf{R}^{d_{1}+d_{3}})$, and

$\displaystyle\|\operatorname{Tr}_{z}f\|_{M^{p_{0},q_{0}}_{(\omega_{0})}}$
$\displaystyle\lesssim$ $\displaystyle\|f\|_{M^{p,q}_{(\omega)}},\qquad f$
$\displaystyle\in M^{p,q}_{(\omega)}(\mathbf{R}^{d_{1}+d_{2}+d_{3}})$ (2.12)

  2. (2)

if in addition (2.8) holds true for some $r_{0}\geq 0$, then the map in (1) a
surjective.

For modulation spaces of Wiener-amalgam types we also have the following. Here
the conditions (2.6) and (2.7) are relaxed into

$\max\left(\frac{1}{q_{1}},\frac{1}{q_{2}},1\right)-\frac{1}{q_{2}}\leq\frac{1}{r},$
(2.13)

and

$\sup_{(x,\xi_{3})\in\mathbf{R}^{d+d_{3}}}\left(\left\|\sup_{\xi_{1}\in\mathbf{R}^{d_{1}}}\left(\frac{\omega_{0}(x_{1},x_{3},\xi_{1},\xi_{3})e^{-r_{0}|x_{2}|}}{\omega(x_{1},x_{2},x_{3},\xi_{1},\,\cdot\,,\xi_{3})}\right)\right\|_{L^{r}(\mathbf{R}^{d_{2}})}\right)<\infty.$
(2.7)′

######  Theorem 2.3.

Let $d_{j}\geq 0$ be integers, $z\in\mathbf{R}^{d_{2}}$ be fixed,
$p_{j},q_{j}\in(0,\infty]^{d_{j}}$, $r\in(0,\infty]^{d_{2}}$, $j=1,2,3$, be
such that (2.13) holds and let
$\omega\in\mathscr{P}_{E}(\mathbf{R}^{2(d_{1}+d_{2}+d_{3})})$,
$\omega_{0}\in\mathscr{P}_{E}(\mathbf{R}^{2(d_{1}+d_{3})})$, $p$, $p_{0}$, $q$
and $q_{0}$ be such that (2.5) and (2.7)′ hold true for some $r_{0}\geq 0$.
Then the following is true:

  1. (1)

the map $\operatorname{Tr}_{z}$ from
$\Sigma_{1}(\mathbf{R}^{d_{1}+d_{2}+d_{3}})$ to
$\Sigma_{1}(\mathbf{R}^{d_{1}+d_{3}})$ extends uniquely to a continuous map
from $W^{p,q}_{(\omega)}(\mathbf{R}^{d_{1}+d_{2}+d_{3}})$ to
$W^{p_{0},q_{0}}_{(\omega_{0})}(\mathbf{R}^{d_{1}+d_{3}})$, and

$\displaystyle\|\operatorname{Tr}_{z}f\|_{W^{p_{0},q_{0}}_{(\omega_{0})}}$
$\displaystyle\lesssim$ $\displaystyle\|f\|_{W^{p,q}_{(\omega)}},\qquad f$
$\displaystyle\in W^{p,q}_{(\omega)}(\mathbf{R}^{d_{1}+d_{2}+d_{3}})\text{;}$
(2.14)

  2. (2)

if in addition (2.8) holds true for some $r_{0}\geq 0$, then the map in (1) is
surjective.

For the proof of Theorems 2.2 and 2.3 we recall the Young type inequality

$\|f_{1}*f_{2}\|_{\ell^{p}_{(\omega)}}\leq\|f_{1}\|_{\ell^{p}_{(\omega)}}\|f_{2}\|_{\ell^{\min(p,1)}_{(v)}},\qquad
f_{1}\in\ell^{p}_{(\omega)}(\varepsilon\mathbf{Z}^{d}),\
f_{2}\in\ell^{\min(p,1)}_{(\omega)}(\varepsilon\mathbf{Z}^{d})$ (2.15)

for discrete Lebesgue spaces, when
$\omega,v\in\mathscr{P}_{E}(\mathbf{R}^{d})$ satisfy
$\omega(x+y)\leq\omega(x)v(y)$. This gives

$\|f*e^{-r|\,\cdot\,|}\|_{\ell^{p}_{(\omega)}(\varepsilon\mathbf{Z}^{d})}\lesssim\|f\|_{\ell^{p}_{(\omega)}(\varepsilon\mathbf{Z}^{d})},\qquad
f\in\ell^{p}_{(\omega)}(\varepsilon\mathbf{Z}^{d}),$ (2.16)

provided $r>0$ is chosen large enough.

######  Remark 2.4.

Let $p\in(0,1]$, $\mathscr{B}$ be a vector space and let
$\|\,\cdot\,\|_{\mathscr{B}}$ be a $p$-norm, i. e.
$\|\,\cdot\,\|_{\mathscr{B}}$ is a quasi-norm on $\mathscr{B}$ which fulfills

$\displaystyle\|f+g\|_{\mathscr{B}}^{p}$
$\displaystyle\leq\|f\|_{\mathscr{B}}^{p}+\|g\|_{\mathscr{B}}^{p},$
$\displaystyle\qquad f,g$ $\displaystyle\in\mathscr{B}.$ Then it follows that
$\displaystyle\|f+g\|_{\mathscr{B}}^{q}$
$\displaystyle\leq\|f\|_{\mathscr{B}}^{q}+\|g\|_{\mathscr{B}}^{q},$
$\displaystyle\qquad f,g$ $\displaystyle\in\mathscr{B},$

for every $q\in(0,p]$.

###### Proof of Theorem 2.2.

First we prove (1). Let $d=d_{1}+d_{2}+d_{3}$, $d_{0}=d_{1}+d_{3}$,
$\rho=\min(p_{0},q_{1},q_{2},1)$, $\rho_{0}=\min(p,q,1)=\min(\rho,q_{3})$ and
$g_{z}=\operatorname{Tr}_{z}f$ and $v\in\mathscr{P}_{E}(\mathbf{R}^{2d})$ be
such that $\omega$ is $v$-moderate. Also let $x$, $\xi$, $x_{0}$, $\xi_{0}$
and $\vartheta$ be given by (2.9) and (2.11). Then
$\vartheta\in\mathscr{P}_{E}(\mathbf{R}^{d_{2}+d_{3}})$ fulfills (2.10).

First suppose that $f\in\Sigma_{1}(\mathbf{R}^{d})$. By Proposition 1.6, there
are $\phi_{m}\in M^{\rho_{0}}_{(v)}(\mathbf{R}^{d})$,
$\psi_{m}\in\Sigma_{1}(\mathbf{R}^{d_{m}})$, $m=1,2,3$ such that

$\displaystyle f(x)$
$\displaystyle=c_{\varepsilon}\sum_{j,\iota\in\varepsilon\mathbf{Z}^{d}}V_{\phi}f(j,\iota)e^{i\langle
x,\iota\rangle}\psi(x-j),$ where $\displaystyle\phi$
$\displaystyle=\phi_{1}\otimes\phi_{2}\otimes\phi_{3}\qquad\text{and}\qquad\psi=\psi_{1}\otimes\psi_{2}\otimes\psi_{3}.$
We have $\displaystyle g_{z}(x_{1},x_{3})$
$\displaystyle=c_{\varepsilon}\sum_{j,\iota\in\varepsilon\mathbf{Z}^{d}}V_{\phi}f(j,\iota)e^{i(\langle
x_{1},\iota_{1}\rangle+\langle z,\iota_{2}\rangle+\langle
x_{3},\iota_{3}\rangle)}\psi(x_{1}-j_{1},z-j_{2},x_{3}-j_{3}),$ (2.17) when
$\displaystyle g_{z}(x_{1},x_{3})$ $\displaystyle=f(x_{1},z,x_{3}),$ (2.18)

and observe that the series possess strong convergence properties, because

$|V_{\phi}f(j,\iota)|\lesssim
e^{-r_{1}(|j|+|\iota|)}\quad\text{and}\quad|\psi(x_{1}-j_{1},z-j_{2},x_{3}-j_{3})|\lesssim
e^{-r_{1}(|x_{1}-j_{1}|+|j_{2}|+|x_{3}-j_{3}|)}$

for every $r_{1}>0$.

An application of the short-time Fourier transform on (2.17) gives

$V_{\psi_{0}}g_{z}(x_{0},\xi_{0})\\\\[4.30554pt]
=c_{\varepsilon}\sum_{j,\iota\in\varepsilon\mathbf{Z}^{d}}V_{\phi}f(j,\iota)e^{i(\langle
z,\iota_{2}\rangle+\langle
j_{0},\iota_{0}-\xi_{0}\rangle)}(V_{\psi_{0}}\psi_{0})(x_{0}-j_{0},\xi_{0}-\iota_{0})\psi_{2}(z-j_{2}),$
(2.19)

where

$j_{0}=(j_{1},j_{3}),\quad\iota_{0}=(\iota_{1},\iota_{3})\quad\text{and}\quad\psi_{0}=\psi_{1}\otimes\psi_{3}.$

By letting

$F_{\omega}=|V_{\phi}f\cdot\omega|\quad\text{and}\quad
F_{0,\omega_{0}}=|V_{\psi_{0}}g_{z}\cdot\omega_{0}|,$

and using that

$\omega_{0}(x_{0},\xi_{0})\lesssim\omega(x_{1},j_{2},x_{3},\xi_{1},\iota_{2},\xi_{3})e^{r_{0}|j_{2}|}\vartheta(\iota_{2},\lambda_{3})\\\
\lesssim\omega(j,\iota)e^{r_{0}(|x_{0}-j_{0}|+|j_{2}|+|\xi_{0}-\iota_{0}|)}\vartheta(\iota_{2},\lambda_{3}).$

and

$|(V_{\psi_{0}}\psi_{0})(x_{0}-j_{0},\xi_{0}-\iota_{0})\psi_{2}(z-j_{2})e^{r_{0}(|x_{0}-j_{0}|+|j_{2}|+|\xi_{0}-\iota_{0}|)}|\lesssim
e^{-r_{1}(|x_{0}-j_{0}|+|j_{2}|+|\xi_{0}-\iota_{0}|)}$

for every $r_{1}>0$, it follows from (2.19) that

$F_{0,\omega_{0}}(l_{0},\lambda_{0})\lesssim\sum_{j,\iota\in\varepsilon\mathbf{Z}^{d}}F_{\omega}(j,\iota)e^{-r_{1}(|l_{0}-j_{0}|+|j_{2}|+|\lambda_{0}-\iota_{0}|)}\vartheta(\iota_{2},\lambda_{3}),\qquad
r_{1}>0,$ (2.20)

where

$l_{0}=(l_{1},l_{3})\in\varepsilon\mathbf{Z}^{d_{0}}\quad\text{and}\quad\lambda_{0}=(\lambda_{1},\lambda_{3})\in\varepsilon\mathbf{Z}^{d_{0}}.$

Since $\rho\in(0,1]$, (2.20) gives

$F_{0,\omega_{0}}(l_{0},\lambda_{0})^{\rho}\lesssim\sum_{j,\iota\in\varepsilon\mathbf{Z}^{d}}F_{\omega}(j,\iota)^{\rho}e^{-r_{1}(|l_{0}-j_{0}|+|j_{2}|+|\lambda_{0}-\iota_{0}|)}\vartheta(\iota_{2},\lambda_{3})^{\rho},\qquad
r_{1}>0.$ (2.20)′

Now let

$\displaystyle\Lambda_{1}=\varepsilon\mathbf{Z}^{d_{2}+d_{3}+d},\quad\Lambda_{2}=\varepsilon\mathbf{Z}^{d_{3}+d},\quad\Lambda_{3}=\varepsilon\mathbf{Z}^{d_{2}+d_{3}},$
$\displaystyle
F_{p_{1},\omega}(x_{2},x_{3},\xi)=\|F_{\omega}(\,\cdot\,,x_{2},x_{3},\xi)\|_{\ell^{p_{1}}(\varepsilon\mathbf{Z}^{d_{1}})},\quad
F_{p_{1},p_{2},\omega}(x_{3},\xi)=\|F_{\omega}(\,\cdot\,,x_{3},\xi)\|_{\ell^{p_{1},p_{2}}(\varepsilon\mathbf{Z}^{d_{1}+d_{2}})},$
$\displaystyle
F_{p,\omega}(\xi)=\|F_{\omega}(\,\cdot\,,\xi)\|_{\ell^{p}(\varepsilon\mathbf{Z}^{d})},\quad
F_{p,q_{1},\omega}(\xi_{2},\xi_{3})=\|F_{\omega}(\,\cdot\,,\xi_{2},\xi_{3})\|_{\ell^{p,q_{1}}(\varepsilon\mathbf{Z}^{d+d_{1}})},$
$\displaystyle
F_{p,q_{1}q_{2},\omega}(\xi_{3})=\|F_{\omega}(\,\cdot\,,\xi_{3})\|_{\ell^{p,q_{1},q_{2}}(\varepsilon\mathbf{Z}^{d+d_{1}+d_{2}})}$
and $\displaystyle
G_{r_{1},\rho,\omega}(l_{1},j_{2},j_{3},\iota)=(F_{\omega}(\,\cdot\,,j_{2},j_{3},\iota)^{\rho}*e^{-r_{1}|\,\cdot\,|})(l_{1}),$

where the convolution is the discrete convolution with respect to
$\varepsilon\mathbf{Z}^{d_{1}}$. By (2.20)′ it follows that

$F_{0,\omega_{0}}(l_{0},\lambda_{0})^{\rho}\lesssim\sum_{(j_{2},j_{3},\iota)\in\Lambda_{1}}(F_{\omega}(\,\cdot\,,j_{2},j_{3},\iota)^{\rho}*e^{-r_{1}|\,\cdot\,|})(l_{1})e^{-r_{1}(|j_{2}|+|l_{3}-j_{3}|+|\lambda_{0}-\iota_{0}|)}\vartheta(\iota_{2},\lambda_{3})^{\rho},$

for every $r_{1}>0$. If we apply the $\ell^{p_{1}/\rho}$ norm with respect to
$l_{1}$ variable, and use Minkowski’s, Young’s and Hölder’s inequalities, we
obtain

$\|F_{0,\omega_{0}}(\,\cdot\,,l_{3},\lambda_{0})^{\rho}\|_{\ell^{p_{1}/\rho}}=\|F_{0,\omega_{0}}(\,\cdot\,,l_{3},\lambda_{0})\|_{\ell^{p_{1}}}^{\rho}\\\\[4.30554pt]
\lesssim\left\|\sum_{(j_{2},j_{3},\iota)\in\Lambda_{1}}G_{r_{1},\rho,\omega}(\,\cdot\,,j_{2},j_{3},\iota)e^{-r_{1}(|j_{2}|+|l_{3}-j_{3}|+|\lambda_{0}-\iota_{0}|)}\vartheta(\iota_{2},\lambda_{3})^{\rho}\right\|_{\ell^{p_{1}/\rho}}\\\\[4.30554pt]
\lesssim\sum_{(j_{2},j_{3},\iota)\in\Lambda_{1}}\left(\|F_{\omega}(\,\cdot\,,j_{2},j_{3},\iota)^{\rho}*e^{-r_{1}|\,\cdot\,|}\|_{\ell^{p_{1}/\rho}}\right)e^{-r_{1}(|j_{2}|+|l_{3}-j_{3}|+|\lambda_{0}-\iota_{0}|)}\vartheta(\iota_{2},\lambda_{3})^{\rho}\\\\[4.30554pt]
\lesssim\sum_{(j_{2},j_{3},\iota)\in\Lambda_{1}}\left(\|F_{\omega}(\,\cdot\,,j_{2},j_{3},\iota)^{\rho}\|_{\ell^{p_{1}/\rho}}\right)e^{-r_{1}(|j_{2}|+|l_{3}-j_{3}|+|\lambda_{0}-\iota_{0}|)}\vartheta(\iota_{2},\lambda_{3})^{\rho}\\\\[4.30554pt]
=\sum_{(j_{2},j_{3},\iota)\in\Lambda_{1}}F_{p_{1},\omega}(j_{2},j_{3},\iota)^{\rho}e^{-r_{1}(|j_{2}|+|l_{3}-j_{3}|+|\lambda_{0}-\iota_{0}|)}\vartheta(\iota_{2},\lambda_{3})^{\rho}\\\\[4.30554pt]
\leq\sum_{(j_{3},\iota)\in\Lambda_{2}}\|F_{p_{1},\omega}(\,\cdot\,,j_{3},\iota)^{\rho}\|_{\ell^{\infty}}\|e^{-r_{1}|\,\cdot\,|}\|_{\ell^{1}}e^{-r_{1}(|l_{3}-j_{3}|+|\lambda_{0}-\iota_{0}|)}\vartheta(\iota_{2},\lambda_{3})^{\rho}\\\\[4.30554pt]
\lesssim\sum_{(j_{3},\iota)\in\Lambda_{2}}F_{p_{1},p_{2},\omega}(j_{3},\iota)^{\rho}e^{-r_{1}(|l_{3}-j_{3}|+|\lambda_{0}-\iota_{0}|)}\vartheta(\iota_{2},\lambda_{3})^{\rho},$

for every $r_{1}>0$. In the last inequality we have used the fact that
$\|f\|_{\ell^{\infty}(\varepsilon\mathbf{Z}^{d_{2}})}\leq\|f\|_{\ell^{p}(\varepsilon\mathbf{Z}^{d_{2}})}$
for every sequence $f$ on $\varepsilon\mathbf{Z}^{d_{2}}$ and
$p\in(0,\infty]$. Hence, if

$G_{p_{1},p_{2},r_{1},\rho,\omega}(l_{3},\iota)=(F_{p_{1},p_{2},\omega}(\,\cdot\,,\iota)^{\rho}*e^{-r_{1}|\,\cdot\,|})(l_{3}),$

where the convolution is the discrete convolution with respect to
$\varepsilon\mathbf{Z}^{d_{3}}$, we get

$\|F_{0,\omega_{0}}(\,\cdot\,,l_{3},\lambda_{0})\|_{\ell^{p_{1}}}^{\rho}\lesssim\sum_{\iota\in\varepsilon\mathbf{Z}^{d}}G_{p_{1},p_{2},r_{1},\rho,\omega}(l_{3},\iota)e^{-r_{1}|\lambda_{0}-\iota_{0}|}\vartheta(\iota_{2},\lambda_{3})^{\rho},\quad
r_{1}>0.$

An application of the $\ell^{p_{3}/\rho}$ norm with respect to $l_{3}$
variable, and using Minkowski’s, Young’s and Hölder’s inequalities now give

$\|F_{0,\omega_{0}}(\,\cdot\,,\lambda_{0})\|_{\ell^{p_{0}}}^{\rho}\lesssim\left\|\sum_{\iota\in\varepsilon\mathbf{Z}^{d}}G_{p_{1},p_{2},r_{1},\rho,\omega}(\,\cdot\,,\iota)e^{-r_{1}|\lambda_{0}-\iota_{0}|}\vartheta(\iota_{2},\lambda_{3})^{\rho}\right\|_{\ell^{p_{3}/\rho}}\\\\[4.30554pt]
\lesssim\sum_{\iota\in\varepsilon\mathbf{Z}^{d}}\left(\|F_{p_{1},p_{2},\omega}(\,\cdot\,,\iota)^{\rho}*e^{-r_{1}|\,\cdot\,|}\|_{\ell^{p_{3}/\rho}}\right)e^{-r_{1}|\lambda_{0}-\iota_{0}|}\vartheta(\iota_{2},\lambda_{3})^{\rho}\\\\[4.30554pt]
\lesssim\sum_{\iota\in\varepsilon\mathbf{Z}^{d}}\|F_{p_{1},p_{2},\omega}(\,\cdot\,,\iota)^{\rho}\|_{\ell^{p_{3}/\rho}}e^{-r_{1}|\lambda_{0}-\iota_{0}|}\vartheta(\iota_{2},\lambda_{3})^{\rho}\\\\[4.30554pt]
=\sum_{\iota\in\varepsilon\mathbf{Z}^{d}}F_{p,\omega}(\iota)^{\rho}e^{-r_{1}|\lambda_{0}-\iota_{0}|}\vartheta(\iota_{2},\lambda_{3})^{\rho}$

for every $r_{1}>0$. That is

$\|F_{0,\omega_{0}}(\,\cdot\,,\lambda_{0})\|_{\ell^{p_{0}}}^{\rho}\lesssim\sum_{(\iota_{2},\iota_{3})\in\Lambda_{3}}G_{p,r_{1},\rho,\omega}(\lambda_{1},\iota_{2},\iota_{3})e^{-r_{1}|\lambda_{3}-\iota_{3}|}\vartheta(\iota_{2},\lambda_{3})^{\rho},\quad
r_{1}>0,$

where

$G_{p,r_{1},\rho,\omega}(\iota)=(F_{p,\omega}(\,\cdot\,,\iota_{2},\iota_{3})^{\rho}*e^{-r_{1}|\,\cdot\,|})(\iota_{1})$

and the convolution is the discrete convolution with respect to
$\varepsilon\mathbf{Z}^{d_{1}}$. If we apply the $\ell^{q_{1}/\rho}$ norm with
respect to $\lambda_{1}$ variable, and use Minkowski’s, Young’s and Hölder’s
inequalities, we obtain

$\|F_{0,\omega_{0}}(\,\cdot\,,\lambda_{3})\|_{\ell^{p_{0},q_{1}}}^{\rho}\lesssim\left\|\sum_{(\iota_{2},\iota_{3})\in\Lambda_{3}}G_{p,r_{1},\rho,\omega}(\lambda_{1},\iota_{2},\iota_{3})e^{-r_{1}|\lambda_{3}-\iota_{3}|}\vartheta(\iota_{2},\lambda_{3})^{\rho}\right\|_{\ell^{q_{1}/\rho}}\\\\[4.30554pt]
\leq\sum_{(\iota_{2},\iota_{3})\in\Lambda_{3}}\|F_{p,\omega}(\,\cdot\,,\iota_{2},\iota_{3})^{\rho}*e^{-r_{1}|\,\cdot\,|}\|_{\ell^{q_{1}/\rho}}e^{-r_{1}|\lambda_{3}-\iota_{3}|}\vartheta(\iota_{2},\lambda_{3})^{\rho}\\\\[4.30554pt]
\lesssim\sum_{(\iota_{2},\iota_{3})\in\Lambda_{3}}\|F_{p,\omega}(\,\cdot\,,\iota_{2},\iota_{3})^{\rho}\|_{\ell^{q_{1}/\rho}}e^{-r_{1}|\lambda_{3}-\iota_{3}|}\vartheta(\iota_{2},\lambda_{3})^{\rho}\\\\[4.30554pt]
=\sum_{(\iota_{2},\iota_{3})\in\Lambda_{3}}F_{p,q_{1},\omega}(\iota_{2},\iota_{3})^{\rho}e^{-r_{1}|\lambda_{3}-\iota_{3}|}\vartheta(\iota_{2},\lambda_{3})^{\rho}\\\\[4.30554pt]
\leq\sum_{\iota_{3}\in\varepsilon\mathbf{Z}^{d_{3}}}\|F_{p,q_{1},\omega}(\,\cdot\,,\iota_{3})^{\rho}\|_{\ell^{q_{2}/\rho}}\|\vartheta(\,\cdot\,,\lambda_{3})^{\rho}\|_{\ell^{(q_{2}/\rho)^{\prime}}}e^{-r_{1}|\lambda_{3}-\iota_{3}|}\\\\[4.30554pt]
=\sum_{\iota_{3}\in\varepsilon\mathbf{Z}^{d_{3}}}\left(F_{p,q_{1},q_{2},\omega}(\iota_{3})\|\vartheta(\,\cdot\,,\lambda_{3})\|_{\ell^{r}}\right)^{\rho}e^{-r_{1}|\lambda_{3}-\iota_{3}|}\\\\[4.30554pt]
\leq
C_{\vartheta}\sum_{\iota_{3}\in\varepsilon\mathbf{Z}^{d_{3}}}F_{p,q_{1},q_{2},\omega}(\iota_{3})^{\rho}e^{-r_{1}|\lambda_{3}-\iota_{3}|}\asymp(F_{p,q_{1},q_{2},\omega}^{\rho}*e^{-r_{1}|\,\cdot\,|})(\lambda_{3})$

for every $r_{1}>0$. Here $C_{\vartheta}$ is given by (2.10).

By applying the $\ell^{q_{3}/\rho}$ quasi-norm on the latter inequality we
obtain

$\|F_{0,\omega_{0}}\|_{\ell^{p_{0},q_{0}}(\varepsilon\mathbf{Z}^{2d_{0}})}^{\rho}\lesssim\left\|F_{p,q_{1},q_{2},\omega}^{\rho}*e^{-r_{1}|\,\cdot\,|}\right\|_{\ell^{q_{3}/\rho}(\varepsilon\mathbf{Z}^{d_{3}})}\\\\[4.30554pt]
\leq\|F_{p,q_{1},q_{2},\omega}^{\rho}\|_{\ell^{q_{3}/\rho}(\varepsilon\mathbf{Z}^{d_{3}})}\|e^{-r_{1}|\,\cdot\,|}\|_{\ell^{\min(1,q_{3}/\rho)}(\varepsilon\mathbf{Z}^{d_{3}})}$

for every $r_{1}>0$, which gives

$\|F_{0,\omega_{0}}\|_{\ell^{p_{0},q_{0}}(\varepsilon\mathbf{Z}^{2d_{0}})}\lesssim\|F_{p,q_{1},q_{2},\omega}\|_{\ell^{q_{3}}(\varepsilon\mathbf{Z}^{d_{3}})}=\|F_{\omega}\|_{\ell^{p,q}(\varepsilon\mathbf{Z}^{2d})}.$

This gives (2.12) for $f\in\Sigma(\mathbf{R}^{d_{1}+d_{2}+d_{3}})$, i. e.

$\|\operatorname{Tr}_{z}f\|_{M^{p_{0},q_{0}}_{(\omega_{0})}}\lesssim\|f\|_{M^{p,q}_{(\omega)}},\qquad
f\in\Sigma_{1}(\mathbf{R}^{d_{1}+d_{2}+d_{3}})$ (2.12)′

The assertion now follows in the case $\max(p,q)<\infty$ from (2.12)′ and the
fact that $\Sigma_{1}(\mathbf{R}^{d_{1}+d_{2}+d_{3}})$ is dense in
$M^{p,q}_{(\omega)}(\mathbf{R}^{d_{1}+d_{2}+d_{3}})$ in view of (1) in
Proposition 1.5.

For the case $\max(p,q)=\infty$, we first assume that $\min(p,q)\geq 1$. Then
all involved modulation spaces are Banach spaces, and it follows by (2.12)′
and Hahn-Banach’s theorem that $\operatorname{Tr}_{z}$ extends to a continuous
map from $M^{p,q}_{(\omega)}(\mathbf{R}^{d_{1}+d_{2}+d_{3}})$ to
$M^{p_{0},q_{0}}_{(\omega_{0})}(\mathbf{R}^{d_{1}+d_{3}})$.

In order to show that this extension is unique, let

$\displaystyle\omega_{0,1}(x_{0},\xi_{0})$
$\displaystyle=\omega_{0}(x_{0},\xi_{0})e^{-r_{0}(|x_{0}|+|\xi_{0}|)}$ and
$\displaystyle\omega_{1}(x,\xi)$
$\displaystyle=\omega(x,\xi)e^{-r_{0}(|x|+|\xi_{0}|)}\vartheta(\xi_{2},\xi_{3}).$

Then

$M^{p,q}_{(\omega)}(\mathbf{R}^{d})\hookrightarrow
M^{1,1}_{(\omega_{1})}(\mathbf{R}^{d}),\qquad
M^{p_{0},q_{0}}_{(\omega_{0})}(\mathbf{R}^{d_{0}})\hookrightarrow
M^{1,1}_{(\omega_{0,1})}(\mathbf{R}^{d_{0}}),$ (2.21)

and (2.7) implies

$\sup_{\xi_{3}}\left(\left\|\sup_{x,\xi_{1}}\left(\frac{\omega_{0,1}(x_{0},\xi_{0})e^{-r_{0}|x_{2}|}}{\omega(x,\xi_{1},\,\cdot\,,\xi_{3})}\right)\right\|_{L^{\infty}(\mathbf{R}^{d_{2}})}\right)<\infty$

(see also (0.10)). Since the assertion holds in the case $\max(p,q)<\infty$,
it now follows that $\operatorname{Tr}_{z}$ is uniquely defined and continuous
from $M^{1,1}_{(\omega_{1})}(\mathbf{R}^{d})$ to
$M^{1,1}_{(\omega_{0,1})}(\mathbf{R}^{d_{0}})$. In particular, if $f\in
M^{p,q}_{(\omega)}(\mathbf{R}^{d})$, then $\operatorname{Tr}_{z}f$ is uniquely
defined as an element in $M^{1,1}_{(\omega_{0,1})}(\mathbf{R}^{d_{0}})$ due to
the inclusions above. The asserted uniqueness now follows from the fact that
if $g_{1},g_{2}\in M^{p_{0},q_{0}}_{(\omega_{0})}(\mathbf{R}^{d_{0}})$ and
$g_{1}=g_{2}$ as elements in $M^{1,1}_{(\omega_{0,1})}(\mathbf{R}^{d_{0}})$,
then $g_{1}=g_{2}$ as elements in
$M^{p_{0},q_{0}}_{(\omega_{0})}(\mathbf{R}^{d_{0}})$.

Finally suppose that $p$ and $q$ are general, and let $\omega_{1}$ and
$\omega_{0,1}$ be as above. Since (2.21) holds when $\min(p,q)\geq 1$ and that
$M^{p,q}_{(\omega)}(\mathbf{R}^{d})$ increases with $p$ and $q$, it follows
that (2.21) still holds without the restriction $\min(p,q)\geq 1$. Hence, if
$f\in M^{p,q}_{(\omega)}(\mathbf{R}^{d})$, then $\operatorname{Tr}_{z}f$ is
uniquely defined as an element in
$M^{1,1}_{(\omega_{0,1})}(\mathbf{R}^{d_{0}})$. By the computations which lead
to (2.12)′, it follows that $\operatorname{Tr}_{z}f\in
M^{p_{0},q_{0}}_{(\omega_{0})}(\mathbf{R}^{d_{0}})$, and that (2.12) holds.
This gives (1).

It remains to prove (2). Let $f_{0}\in
M^{p,q}_{(\omega_{0})}(\mathbf{R}^{d_{0}})$ be fixed, $\phi_{j}$ and $\phi$ be
as above, $\varphi\in\Sigma_{1}(\mathbf{R}^{d_{2}})$ be such that
$\varphi(0)=1$, and let

$f(x_{1},x_{2},x_{3})=f_{0}(x_{1},x_{3})\varphi(x_{2}-z).$

We have

$V_{\phi_{2}}\varphi(x_{2},\xi_{2})\lesssim e^{-2r_{1}(|x_{2}|+|\xi_{2}|)}$

for every $r_{1}>r_{0}$, in view of Lemma 1.1. Hence, by letting

$F=|V_{\phi}f|\cdot\omega,\qquad F_{0}=|V_{\phi_{0}}f_{0}|\cdot\omega_{0}$

and using (2.8), we obtain

$F(x_{1},x_{2},x_{3},\xi_{1},\xi_{2},\xi_{3})=|V_{\phi_{0}}f_{0}(x_{1},x_{3},\xi_{1},\xi_{3})V_{\phi_{2}}\varphi(x_{2}-z,\xi_{2})\omega(x_{1},x_{2},x_{3},\xi_{1},\xi_{2},\xi_{3})|\\\\[4.30554pt]
\lesssim
F_{0}(x_{1},x_{3},\xi_{1},\xi_{3})e^{r_{0}|x_{2}|}e^{-2r(|x_{2}-z|+|\xi_{2}|)}\lesssim
F_{0}(x_{1},x_{3},\xi_{1},\xi_{3})e^{-r_{1}(|x_{2}|+|\xi_{2}|)}.$

An application of the $L^{p,q}$ norm on the latter inequality now gives

$\|f\|_{M^{p,q}_{(\omega)}}\lesssim\|g\|_{L^{p_{2},q_{2}}}\|f_{0}\|_{M^{p_{0},q_{0}}_{(\omega_{0})}}\asymp\|f_{0}\|_{M^{p_{0},q_{0}}_{(\omega_{0})}},\quad
g(x_{2},\xi_{2})=e^{-r_{1}(|x_{2}|+|\xi_{2}|)},$

which in turn implies that $f\in M^{p,q}_{(\omega)}(\mathbf{R}^{3d})$, and the
surjectivity follows. ∎

###### Proof of Theorem 2.3.

The result follows by similar arguments as in the proof of Theorem 2.2. In
order for clarifying some details, we present the first part of the proof.

We use the same notation as in the proof of Theorem 2.2, except that we let
$\rho=\min(q_{1},q_{2},1)$,

$\vartheta(x,\xi_{2},\xi_{3})\equiv\sup_{\xi_{1}\in\mathbf{R}^{d_{1}}}\left(\frac{\omega_{0}(x_{0},\xi_{0})e^{-r_{0}|x_{2}|}}{\omega(x,\xi)}\right)\in\mathscr{P}_{E}(\mathbf{R}^{d+d_{2}+d_{3}}),$
$\displaystyle\Lambda_{1}=\varepsilon\mathbf{Z}^{d+d_{2}+d_{3}},\quad\Lambda_{2}=\varepsilon\mathbf{Z}^{d+d_{3}},\quad\Lambda_{3}=\varepsilon\mathbf{Z}^{d_{2}+d_{3}},$
$\displaystyle
F_{q_{1},\omega}(x,\xi_{2},\xi_{3})=\|F_{\omega}(x,\,\cdot\,,\xi_{2},\xi_{3})\|_{\ell^{q_{1}}(\varepsilon\mathbf{Z}^{d_{1}})},$
and $\displaystyle
G_{r_{1},\rho,\omega}(j,\lambda_{1},\iota_{2},\iota_{3})=(F_{\omega}(j,\,\cdot\,,\iota_{2},\iota_{3})^{\rho}*e^{-r_{1}|\,\cdot\,|})(\lambda_{1}),$

where the convolution is the discrete convolution with respect to
$\varepsilon\mathbf{Z}^{d_{1}}$. Then

$\omega_{0}(x_{0},\xi_{0})e^{-r_{0}|x_{2}|}\leq\vartheta(x,\xi_{2},\xi_{3})\omega(x,\xi),$
(2.22)

and (2.7)′ implies

$C_{\vartheta}=\sup_{x,\xi_{3}}\|\vartheta(x,\,\cdot\,,\xi_{3})\|_{L^{r}}<\infty.$
(2.7)′′

Suppose that $f\in\Sigma_{1}(\mathbf{R}^{d})$. By (2.20)′ and (2.22) it
follows that

$F_{0,\omega_{0}}(l_{0},\lambda_{0})^{\rho}\\\\[4.30554pt]
\lesssim\sum_{(j,\iota_{2},\iota_{3})\in\Lambda_{1}}(F_{\omega}(j,\,\cdot\,\iota_{2},\iota_{3})^{\rho}*e^{-r_{1}|\,\cdot\,|})(\lambda_{1})e^{-r_{1}(|l_{0}-j_{0}|+|j_{2}|+|\lambda_{3}-\iota_{3}|)}\vartheta(j,\iota_{2},\lambda_{3})^{\rho},\quad
r_{1}>0.$

If we apply the $\ell^{q_{1}/\rho}$ norm with respect to $\lambda_{1}$
variable, and use Minkowski’s and Young’s inequalities, we obtain

$\|F_{0,\omega_{0}}(l_{0},\,\cdot\,,\lambda_{3})^{\rho}\|_{\ell^{q_{1}/\rho}}=\|F_{0,\omega_{0}}(l_{0},\,\cdot\,,\lambda_{3})\|_{\ell^{q_{1}}}^{\rho}\\\\[4.30554pt]
\lesssim\left\|\sum_{(j,\iota_{2},\iota_{3})\in\Lambda_{1}}G_{r_{1},\rho,\omega}(j,\,\cdot\,,\iota_{2},\iota_{3})e^{-r_{1}(|l_{0}-j_{0}|+|j_{2}|+|\lambda_{3}-\iota_{3}|)}\vartheta(j,\iota_{2},\lambda_{3})^{\rho}\right\|_{\ell^{q_{1}/\rho}}\\\\[4.30554pt]
\lesssim\sum_{(j,\iota_{2},\iota_{3})\in\Lambda_{1}}\left(\|F_{\omega}(j,\,\cdot\,,\iota_{2},\iota_{3})^{\rho}*e^{-r_{1}|\,\cdot\,|}\|_{\ell^{q_{1}/\rho}}\right)e^{-r_{1}(|l_{0}-j_{0}|+|j_{2}|+|\lambda_{3}-\iota_{3}|)}\vartheta(j,\iota_{2},\lambda_{3})^{\rho}\\\\[4.30554pt]
\lesssim\sum_{(j,\iota_{2},\iota_{3})\in\Lambda_{1}}F_{q_{1},\omega}(j,\iota_{2},\iota_{3})^{\rho}e^{-r_{1}(|l_{0}-j_{0}|+|j_{2}|+|\lambda_{3}-\iota_{3}|)}\vartheta(j,\iota_{2},\lambda_{3})^{\rho},$

for every $r_{1}>0$. By using Hölder’s inequality with respect to the
$\iota_{2}$ variable in the sum we obtain

$\|F_{0,\omega_{0}}(l_{0},\,\cdot\,,\lambda_{3})^{\rho}\|_{\ell^{q_{1}/\rho}}\\\\[4.30554pt]
\lesssim\sum_{(j,\iota_{3})\in\Lambda_{1}}\|F_{q_{1},\omega}(j,\,\cdot\,,\iota_{3})^{\rho}\|_{\ell^{q_{2}/\rho}}e^{-r_{1}(|l_{0}-j_{0}|+|j_{2}|+|\lambda_{3}-\iota_{3}|)}\|\vartheta(j,\,\cdot\,,\lambda_{3})^{\rho}\|_{\ell^{(q_{2}/\rho)^{\prime}}}.\\\\[4.30554pt]
\leq
C_{\vartheta}\sum_{(j,\iota_{3})\in\Lambda_{1}}\|F_{q_{1},\omega}(j,\,\cdot\,,\iota_{3})\|_{\ell^{q_{2}}}^{\rho}e^{-r_{1}(|l_{0}-j_{0}|+|j_{2}|+|\lambda_{3}-\iota_{3}|)}.$

for every $r_{1}>0$. The remaining part of the proof is performed in similar
ways as in the proof of Theorem 2.2, by applying Minkowski’s and Young’s
inequalities in suitable ways. The details are left for the reader. ∎

By letting $\omega$ and $\omega_{0}$ be related as

$\omega_{0}(x_{1},x_{3},\xi_{1},\xi_{3})\lesssim\omega(x_{1},x_{2},x_{3},\xi_{1},\xi_{2},\xi_{3})\vartheta(\xi_{2},\xi_{3}),$
(2.23)

we get the following special case of Theorem 2.2. The details are left for the
reader.

######  Proposition 2.5.

Let $d_{j}$ be integers, $z\in\mathbf{R}^{d_{2}}$ be fixed, $p$, $p_{0}$, $q$,
$q_{0}$ and $r$ be the same as in Theorem 2.2, and suppose that
$\omega\in\mathscr{P}_{E}(\mathbf{R}^{2(d_{1}+d_{2}+d_{3})})$,
$\omega_{0}\in\mathscr{P}_{E}(\mathbf{R}^{2(d_{1}+d_{3})})$ and
$\vartheta\in\mathscr{P}_{E}(\mathbf{R}^{d_{2}+d_{3}})$ satisfy (2.10) and
(2.23). Then $\operatorname{Tr}_{z}$ is continuous from
$M^{p,q}_{(\omega)}(\mathbf{R}^{d_{1}+d_{2}+d_{3}})$ to
$M^{p_{0},q_{0}}_{(\omega_{0})}(\mathbf{R}^{d_{1}+d_{3}})$, and from
$W^{p,q}_{(\omega)}(\mathbf{R}^{d_{1}+d_{2}+d_{3}})$ to
$W^{p_{0},q_{0}}_{(\omega_{0})}(\mathbf{R}^{d_{1}+d_{3}})$.

######  Example 2.6.

An important case of Proposition 2.5 appears by letting $\vartheta$ in (2.23)
be given by

$\vartheta(\xi_{2},\xi_{3})=\langle(\xi_{2},\xi_{3})\rangle^{-s}\langle\xi_{3}\rangle^{s_{0}}.$

Here

$s\geq\frac{d_{2}}{r},\quad s_{0}\leq s-\frac{d_{2}}{r}.$

with the first inequality strict when $r<\infty$. Note that (2.10) is
fulfilled for such $\vartheta$. In fact, for $r<\infty$ we have

$\displaystyle\|\vartheta(\,\cdot\,,\xi_{3})\|_{L^{r}}^{r}$
$\displaystyle=\int_{\mathbf{R}^{d_{2}}}\langle(\xi_{2},\xi_{3})\rangle^{-sr}\langle\xi_{3}\rangle^{s_{0}r}\,d\xi_{2}$
$\displaystyle=\langle\xi_{3}\rangle^{(s_{0}-s)r}\int_{\mathbf{R}^{d_{2}}}\langle\xi_{2}/\langle\xi_{3}\rangle\rangle^{-sr}\,d\xi_{2}$
$\displaystyle=\langle\xi_{3}\rangle^{(s_{0}-s)r+d_{2}}\int_{\mathbf{R}^{d_{2}}}\langle\xi_{2}\rangle^{-sr}\,d\xi_{2}\lesssim
1,$

where the inequality follows from the assumptions on $s$ and $s_{0}$. By
similar arguments one gets (2.10) in the case when $r=\infty$. The details are
left for the reader.

By choosing $\omega_{0}(x_{0},\xi_{0})=\langle\xi_{3}\rangle^{s_{0}}$,
$\omega(x,\xi)=\langle(\xi_{2},\xi_{3})\rangle^{s}$ and $p=q=2$, we regain
Sobolev’s embedding theorem (see (0.4) in the introduction).

By letting

$d_{j}=d,\quad p_{j,k}=p_{j}\in(0,\infty]\quad\text{and}\quad
q_{j,k}=q_{j}\in(0,\infty],\quad j\in\\{1,2,3\\},\ k\in\\{1,\dots,d\\},$

and replacing the assumption (2.23) by

$\omega_{0}(x_{1},x_{3},\xi_{1},\xi_{3})\asymp\omega(x_{1},x_{2},x_{3},\xi_{1},\xi_{2},\xi_{3})\langle\xi_{2}\rangle^{-\theta},$
(2.24)

Theorems 2.2 and 2.3 give the following.

######  Proposition 2.7.

Let $z\in\mathbf{R}^{d}$ be fixed, $p=(p_{1},p_{2},p_{3})\in(0,\infty]^{3}$,
$q=(q_{1},q_{2},q_{3})\in(0,\infty]^{3}$ and $r\in(0,\infty]$ be such that
(2.6) holds, and suppose that $\omega\in\mathscr{P}_{E}(\mathbf{R}^{6d})$ and
$\omega_{0}\in\mathscr{P}_{E}(\mathbf{R}^{4d})$ satisfy (2.24), for some

$\theta\geq
d\left(\max\left(\frac{1}{p_{1}},\frac{1}{p_{3}},\frac{1}{q_{1}},\frac{1}{q_{2}},\frac{1}{q_{3}}\right)-\frac{1}{q_{2}}\right)$
(2.25)

with strict inequality when

$\min(p_{1},p_{3},q_{1},q_{3},1)<q_{2}.$

Then $\operatorname{Tr}_{z}$ is continuous and surjective
$M^{p,q}_{(\omega)}(\mathbf{R}^{3d})$ to
$M^{p_{0},q_{0}}_{(\omega_{0})}(\mathbf{R}^{2d})$, and from
$W^{p,q}_{(\omega)}(\mathbf{R}^{3d})$ to
$W^{p_{0},q_{0}}_{(\omega_{0})}(\mathbf{R}^{2d})$.

By choosing $p_{1}=p_{2}=p_{3}$ and $q_{1}=q_{2}=q_{3}$ in the previous
proposition, we get the following.

######  Corollary 2.8.

Let $z\in\mathbf{R}^{d}$ be fixed, $p,q,r\in(0,\infty]$ be such that (2.6)
holds, and suppose that $\omega\in\mathscr{P}_{E}(\mathbf{R}^{6d})$ and
$\omega_{0}\in\mathscr{P}_{E}(\mathbf{R}^{4d})$ satisfy (2.24), for some

$\theta\geq
d\left(\max\left(\frac{1}{p},\frac{1}{q}\right)-\frac{1}{q}\right)$ (2.26)

with strict inequality when $\min(p,1)<q$. Then $\operatorname{Tr}_{z}$ is
continuous and surjective $M^{p,q}_{(\omega)}(\mathbf{R}^{3d})$ to
$M^{p,q}_{(\omega_{0})}(\mathbf{R}^{2d})$, and from
$W^{p,q}_{(\omega)}(\mathbf{R}^{3d})$ to
$W^{p,q}_{(\omega_{0})}(\mathbf{R}^{2d})$.

######  Remark 2.9.

Let $p,q,r\in(0,\infty)$ be such that $0<p<q$, $r=\min(1,p,q)$,

$\displaystyle p_{1}$ $\displaystyle=(p,\dots,p)\in(0,\infty]^{d},$
$\displaystyle\quad p_{2}$ $\displaystyle=(p,\dots,p)\in(0,\infty]^{d-1},$
$\displaystyle q_{1}$ $\displaystyle=(q,\dots,q,r)\in(0,\infty]^{d}$
$\displaystyle\quad\text{and}\quad q_{2}$
$\displaystyle=(q,\dots,q)\in(0,\infty]^{d-1}.$

Also let $s\in\mathbf{R}$,

$\omega_{1}(x_{1},\xi_{1})=\langle\xi_{1}\rangle^{s}\quad\text{and}\quad\omega_{2}(x_{2},\xi_{2})=\langle\xi_{2}\rangle^{s},\quad
x_{1},\xi_{1}\in\mathbf{R}^{d},\ x_{2},\xi_{2}\in\mathbf{R}^{d-1}.$

Then it is proved in [21, Theorem 3.1] that the trace map which takes

$(x_{1},\dots,x_{d})\mapsto
f(x_{1},\dots,x_{d})\quad\text{into}\quad(x_{1},\dots,x_{d-1})\mapsto
f(x_{1},\dots,x_{d-1},0)$

is continuous from $M^{p_{1},q_{1}}_{(\omega_{1})}(\mathbf{R}^{d})$ to
$M^{p_{2},q_{2}}_{(\omega_{2})}(\mathbf{R}^{d-1})$.

See also [10, Theorem 2] for similar results of trace operators on Wiener
amalgam type spaces $W^{p,q}_{(\omega)}(\mathbf{R}^{d})$.

We note that Theorem 2.2 is more general compared to the above result in the
sense of relaxing the conditions of the involved weight functions, as well as
including more exponents in the definition of modulation spaces. For example,
we notice that the trace result above only deals with polynomial type weights
of special kinds, and the range of the Lebesgue exponents $p$ and $q$ is
$(0,\infty)$ instead of the full interval $(0,\infty]$, which includes
$\infty$.

On the other hand, [21, Theorem 3.1] is a special case of [21, Theorem 3.3],
which concern trace results for broader classes of $\alpha$-modulation spaces.
Evidently, Theorem 2.2, Propositions 2.5, 2.7 and Corollary 2.8 do not show
any properties for $\alpha$-modulation spaces which are not modulation spaces.
We note however that some arguments in [39] or in the proof of Theorem 2.2,
might be suitable when extending Theorems 3.1 and 3.3 in [21] to allow the
Lebesgue exponents to belong to the full interval $(0,\infty]$. (Cf. [39,
Remark 3.1] and (0.10).)

######  Remark 2.10.

Let $p$, $q$, $\theta$, $\omega_{0}$, $\omega$, $z$ be the same as in
Proposition 2.7. It is clear that Proposition 2.7 is more general compared to
Corollary 2.8. On the other hand, the involved modulation space
$M^{p,\infty,p,q,q,q}_{(\omega)}(\mathbf{R}^{3d})$ in the domain of
$\operatorname{Tr}_{z}$ in Proposition 2.7 is in some sense more complicated
compared to the corresponding space $M^{p,q}_{(\omega)}(\mathbf{R}^{3d})$ in
Corollary 2.8. An interesting question concerns whether Proposition 2.7
involve important situations which are not included in Corollary 2.8.

In this respect, on one hand we have

$\displaystyle f\in M^{p,\infty,p,q,q,q}_{(\omega)}(\mathbf{R}^{3d})\setminus
M^{p,q}_{(\omega)}(\mathbf{R}^{3d})\quad\text{when}\quad f(x,y,\zeta)$
$\displaystyle=g(x,\zeta),\quad x,y,\zeta\in\mathbf{R}^{d},$ (2.27)
$\displaystyle g$ $\displaystyle\in
M^{p,q}_{(\omega_{0})}(\mathbf{R}^{2d})\setminus 0,$

if in addition $p<\infty$. Here $\omega$ and $\omega_{0}$ should satisfy
(2.24) and (2.26). Hence we may identify
$M^{p,q}_{(\omega_{0})}(\mathbf{R}^{2d})$ with a subset of
$M^{p,\infty,p,q,q,q}_{(\omega)}(\mathbf{R}^{3d})$, but not as a subset of
$M^{p,q}_{(\omega)}(\mathbf{R}^{3d})$. By straight-forward computations it
also follows that $f$ and $g$ in (2.27) fulfill $\operatorname{Tr}_{z}f=g$ for
every $z\in\mathbf{R}^{d}$.

On the other hand, if $p,p_{0}\in(0,\infty]$, then Proposition 2.7 shows that

$\displaystyle\operatorname{Tr}_{z}\,$ $\displaystyle:\,$ $\displaystyle
M^{p,p_{0},p,q,q,q}_{(\omega)}(\mathbf{R}^{3d})$ $\displaystyle\to
M^{p,q}_{(\omega_{0})}(\mathbf{R}^{2d})$ (2.28)

is surjective.

##  3\. Continuity for pseudo-differential operators of amplitude type on
modulation spaces

In this section we shall apply Proposition 2.5 to deduce identification
properties between pseudo-differential operators of amplitude types and
standard types, when the symbol and amplitude classes are modulation spaces.
Then we combine this with suitable continuity and compactness results in [50,
51] to deduce continuity, Schatten-von Neumann and nuclearity properties for
pseudo-differential operators of amplitude types.

###  3.1. Pseudo-differential operators with amplitudes in modulation spaces

It is suitable to consider a modified version of the modulation space
$M^{p,q}_{(\omega)}(\mathbf{R}^{d})$ in order to formulate the main result.
Let

$p_{j},q_{j}\in(0,\infty]^{d},\ j\in\\{1,2,3\\},\quad
p=(p_{1},p_{2},p_{3})\in(0,\infty]^{3d},\quad
q=(q_{1},q_{2},q_{3})\in(0,\infty]^{3d},$

$r=\min(1,p,q)$, $\omega\in\mathscr{P}_{E}(\mathbf{R}^{6d})$ and
$\phi\in\Sigma_{1}(\mathbf{R}^{3d})$. Then let

$\displaystyle\|F\|_{\mathcal{L}^{p,q}(\mathbf{R}^{6d})}$
$\displaystyle\equiv\|G_{1}\|_{L^{p,q}(\mathbf{R}^{6d})},$ $\displaystyle\quad
G_{1}(x,y,\zeta,\xi,\eta,z)$ $\displaystyle=F(x,x+y,\zeta,\xi,\eta,z),$ and
$\displaystyle\|F\|_{\mathcal{L}^{p,q}_{*}(\mathbf{R}^{6d})}$
$\displaystyle\equiv\|G_{2}\|_{L^{q,p}(\mathbf{R}^{6d})},$ $\displaystyle\quad
G_{2}(\xi,\eta,z,x,y,\zeta)$ $\displaystyle=F(x,x+y,\zeta,\xi,\eta,z),$

when $F\in L^{r}_{loc}(\mathbf{R}^{6d})$. The modulation spaces

$\mathcal{M}^{p,q}_{(\omega)}(\mathbf{R}^{3d})=\mathcal{M}^{p_{1},p_{2},p_{3},q_{1},q_{2},q_{3}}_{(\omega)}(\mathbf{R}^{3d})\quad\text{and}\quad\mathcal{W}^{p,q}_{(\omega)}(\mathbf{R}^{3d})=\mathcal{W}^{p_{1},p_{2},p_{3},q_{1},q_{2},q_{3}}_{(\omega)}(\mathbf{R}^{3d})$

are the sets of all $a\in\Sigma_{1}^{\prime}(\mathbf{R}^{3d})$ such that

$\|a\|_{\mathcal{M}^{p,q}_{(\omega)}}=\|a\|_{\mathcal{M}^{p_{1},p_{2},p_{3},q_{1},q_{2},q_{3}}_{(\omega)}}\equiv\|V_{\phi}a\cdot\omega\|_{\mathcal{L}^{p,q}}$

respectively

$\|a\|_{\mathcal{W}^{p,q}_{(\omega)}}=\|a\|_{\mathcal{W}^{p_{1},p_{2},p_{3},q_{1},q_{2},q_{3}}_{(\omega)}}\equiv\|V_{\phi}a\cdot\omega\|_{\mathcal{L}^{p,q}_{*}}$

is finite. The spaces $\mathcal{M}^{p,q}_{(\omega)}(\mathbf{R}^{3d})$ and
$\mathcal{M}^{p,q}_{(\omega)}(\mathbf{R}^{3d})$ are equipped with the
topologies supplied by the quasi-norms
$\|\,\cdot\,\|_{\mathcal{M}^{p,q}_{(\omega)}}$ and
$\|\,\cdot\,\|_{\mathcal{W}^{p,q}_{(\omega)}}$, respectively.

As for the classical modulation spaces $M^{p,q}_{(\omega)}(\mathbf{R}^{3d})$,
we put

$\displaystyle\mathcal{M}^{p_{0,1},p_{0,2},p_{0,3},q_{0,1},q_{0,2},q_{0,3}}_{(\omega)}$
$\displaystyle=\mathcal{M}^{p_{1},p_{2},p_{3},q_{1},q_{2},q_{3}}_{(\omega)}$
and
$\displaystyle\mathcal{W}^{p_{0,1},p_{0,2},p_{0,3},q_{0,1},q_{0,2},q_{0,3}}_{(\omega)}$
$\displaystyle=\mathcal{W}^{p_{1},p_{2},p_{3},q_{1},q_{2},q_{3}}_{(\omega)}$

when $p_{0,j},q_{0,j}\in(0,\infty]$,

$p_{j}=(p_{0,j},\dots,p_{0,j})\in(0,\infty]^{d}\quad\text{and}\quad
q_{j}=(p_{0,j},\dots,q_{0,j})\in(0,\infty]^{d},\quad j=1,2,3.$

The following lemma justifies the introduction of
$\mathcal{M}^{p,q}_{(\omega)}(\mathbf{R}^{3d})$.

######  Lemma 3.1.

Let $p_{j},q_{j}\in(0,\infty]^{d}$, $j=1,2,3$, $p=(p_{1},p_{2},p_{3})$,
$q=(q_{1},q_{2},q_{3})$, $\omega\in\mathscr{P}_{E}(\mathbf{R}^{6d})$,

$\omega_{0}(x,y,\zeta,\xi,\eta,z)=\omega(x,x+y,\zeta,\xi-\eta,\eta,z)$

and let $T_{1}$ be the map on $\Sigma_{1}^{\prime}(\mathbf{R}^{3d})$, given by

$(T_{1}a)(x,y,\zeta)=a(x,x+y,\zeta),\qquad
a\in\Sigma_{1}^{\prime}(\mathbf{R}^{3d}).$

Then $T_{1}$ restricts to a homeomorphism from
$\mathcal{M}^{p,q}_{(\omega)}(\mathbf{R}^{3d})$ to
$M^{p,q}_{(\omega_{0})}(\mathbf{R}^{3d})$

###### Proof.

By straight-forward computations we get

$|(V_{T\phi}(Ta))(x,y,\zeta,\xi,\eta,z)|=|V_{\phi}a(x,x+y,\zeta,\xi-\eta,\eta,z)|$

when $a\in\Sigma_{1}^{\prime}(\mathbf{R}^{3d})$ and
$\phi\in\Sigma_{1}(\mathbf{R}^{3d})$. The result now follows by multiplying
the equality with $\omega_{0}$ and then apply the $L^{p,q}$ quasi-norm. ∎

We also need the following lemma. We omit the proof since the result follows
from [48, Proposition 2.8] and its proof.

######  Lemma 3.2.

Let $p$, $q$, $\omega$ be the same as in Lemma 3.1, let

$\omega_{0}(x,y,\zeta,\xi,\eta,z)=\omega(x,y+z,\eta+\zeta,\xi,\eta,z),$

and let $T_{2}$ be the map

$(T_{2}a)(x,y,\zeta,\xi,\eta,z)=(e^{i\langle
D_{\xi},D_{y}\rangle}a)(x,y,\zeta,\xi,\eta,z),\qquad
a\in\Sigma_{1}^{\prime}(\mathbf{R}^{3d}).$

Then $T_{2}$ from $\Sigma_{1}^{\prime}(\mathbf{R}^{3d})$ to
$\Sigma_{1}^{\prime}(\mathbf{R}^{3d})$ restricts to a homeomorphism from
$M^{p,q}_{(\omega)}(\mathbf{R}^{3d})$ to
$M^{p,q}_{(\omega_{0})}(\mathbf{R}^{3d})$.

Let $a\in\Sigma_{1}(\mathbf{R}^{3d})$, $T_{1}$ and $T_{2}$ be as in Lemmas 3.1
and 3.2. Then (1.29) shows that

$a_{0}=(\operatorname{Tr}_{0}\circ T_{2}\circ T_{1})a.$ (1.29)′

For any $a\in\Sigma_{1}^{\prime}(\mathbf{R}^{3d})$ such that the right-hand
side of (1.29)′ makes sense as an element in
$\Sigma_{1}^{\prime}(\mathbf{R}^{2d})$, we define $\operatorname{Op}(a)$ as
the operator $\operatorname{Op}_{0}(a)$. By previous investigations we get the
following. Here the involved weights should satisfy conditions of the form

$\omega_{0}(x,\zeta,\xi,z)\lesssim\omega(x,x+z,\zeta+\eta,\xi-\eta,\eta,z)\vartheta(\eta,z)$
(3.1)

and

$\omega(x,y,\zeta,\xi,\eta,z)\lesssim\omega_{0}(x,\zeta,\xi,z)e^{r_{0}(|y|+|\eta|)}.$
(3.2)

######  Theorem 3.3.

Let $p_{j},q_{j},r\in(0,\infty]^{d}$, $j=1,2,3$,
$\omega\in\mathscr{P}_{E}(\mathbf{R}^{6d})$,
$\omega_{0}\in\mathscr{P}_{E}(\mathbf{R}^{4d})$, $p$, $p_{0}$, $q$ and $q_{0}$
be such that (2.5), (2.6) and (3.1) hold true for some $r_{0}\geq 0$ and
$\vartheta\in\mathscr{P}_{E}(\mathbf{R}^{2d})$ such that (2.10) holds. Then
the following is true:

  1. (1)

if $a\in\mathcal{M}^{p,q}_{(\omega)}(\mathbf{R}^{3d})$, then there is a unique
$a_{0}\in M^{p_{0},q_{0}}_{(\omega_{0})}(\mathbf{R}^{2d})$ such that
$\operatorname{Op}(a)=\operatorname{Op}_{0}(a_{0})$;

  2. (2)

if in addition (3.2) holds true, and $a_{0}\in
M^{p_{0},q_{0}}_{(\omega_{0})}(\mathbf{R}^{2d})$, then there is an
$a\in\mathcal{M}^{p,q}_{(\omega)}(\mathbf{R}^{3d})$ such that
$\operatorname{Op}(a)=\operatorname{Op}_{0}(a_{0})$.

###### Proof.

The assertion (1) follows by combining Theorem 2.2 (1), Lemma 3.1 and Lemma
3.2, and the observation that (3.1) is the same as that

$\omega_{0}(x,\zeta,\xi,z)\lesssim\omega(x,x+y+z,\zeta+\eta,\xi-\eta,\eta,z)e^{r_{0}|y|}\vartheta(\eta,z)$
(3.1)′

should hold for some $r_{0}>0$, due to the moderateness of $\omega$. The
details are left for the reader.

The assertion (2) follows by letting

$a(x,y,\zeta)=e^{-i\langle D_{\xi},D_{y}\rangle}(a_{0}(x,\zeta)\varphi(y-x)),$

where $\varphi\in\Sigma_{1}(\mathbf{R}^{d})$ fulfills $\varphi(0)=1$, and
applying Theorem 2.2 (2), Lemma 3.1 and Lemma 3.2. The details are left for
the reader. ∎

In a similar way as when passing from Theorem 2.2 into Proposition 2.5, it
follows that the following result is a special cases of Theorem 3.3. The
details are left for the reader. Here the involved weights are related as

$\omega_{0}(x,\zeta,\xi,z)\asymp\omega(x,x+z,\zeta+\eta,\xi-\eta,\eta,z)\vartheta(\eta,z)$
(3.3)

######  Proposition 3.4.

Let $p$, $p_{0}$, $q$, $q_{0}$, $r$ and $\vartheta$ be the same as in Theorem
3.3, and suppose that $\omega\in\mathscr{P}_{E}(\mathbf{R}^{6d})$ and
$\omega_{0}\in\mathscr{P}_{E}(\mathbf{R}^{4d})$ satisfy (3.3). Then

$\\{\,\operatorname{Op}(a)\,;\,a\in\mathcal{M}^{p,q}_{(\omega)}(\mathbf{R}^{3d})\,\\}=\\{\,\operatorname{Op}_{0}(a_{0})\,;\,a_{0}\in
M^{p,q}_{(\omega)}(\mathbf{R}^{2d})\,\\}.$

In the same way it follows that the next result is a special case of the
previous one. The details are left for the reader. Here the relationship
between the weights are more specified into

$\omega_{0}(x,\zeta,\xi,z)\asymp\omega(x,x+z,\zeta+\eta,\xi-\eta,\eta,z)\langle\eta\rangle^{-\theta}$
(3.3)′

######  Proposition 3.5.

Let $p,q\in(0,\infty]$, and suppose that
$\omega\in\mathscr{P}_{E}(\mathbf{R}^{6d})$ and
$\omega_{0}\in\mathscr{P}_{E}(\mathbf{R}^{4d})$ satisfy (3.3)′, for some
$\theta$ such that (2.26) holds with strict inequality when $q>\min(p,1)$.
Then

$\\{\,\operatorname{Op}(a)\,;\,a\in\mathcal{M}^{p,q}_{(\omega)}(\mathbf{R}^{3d})\,\\}=\\{\,\operatorname{Op}_{0}(a_{0})\,;\,a_{0}\in
M^{p,q}_{(\omega_{0})}(\mathbf{R}^{2d})\,\\}.$

We observe that
$\mathcal{M}^{p,q}_{(\omega)}(\mathbf{R}^{3d})=M^{p,q}_{(\omega)}(\mathbf{R}^{3d})$
in the last proposition.

We may combine the previous results with other established results for pseudo-
differential operators when acting on modulation spaces. For example, the
following result is a straight-forward consequence of Theorem 3.3 or
Proposition 3.4 with Theorem 1.8. The details are left for the reader. Here
the condition (0.12) from the introduction needs to be modified into

$\displaystyle\frac{\omega_{2}(x,\xi)}{\omega_{1}(z,\zeta)}$
$\displaystyle\lesssim\omega(x,z,\zeta+\eta,\xi-\zeta-\eta,\eta,z-x)\vartheta(\eta,z),$
$\displaystyle\qquad x,z,\xi,\eta,\zeta$ $\displaystyle\in\mathbf{R}^{d}.$
(0.12)′

Recall also Subsection 1.4 for ordering relations between elements in
$(0,\infty]$ and elements in $(0,\infty]^{d}$.

######  Theorem 3.6.

Let $\sigma\in\operatorname{S}_{2d}$, $p,q,r\in(0,\infty]$ and
$p_{1},p_{2}\in(0,\infty]^{2d}$ be such that

$\frac{1}{p_{2}}-\frac{1}{p_{1}}=\frac{1}{p}+\min\left(0,\frac{1}{q}-1\right),\quad
q\leq\min(p_{2})\leq\max(p_{2})\leq p,$

and

$\max\left(1,\frac{1}{p},\frac{1}{q}\right)-\frac{1}{q}\leq\frac{1}{r},$ (3.4)

hold with latter inequality strict when $q>\min(1,p)$. Also let
$\omega\in\mathscr{P}_{E}(\mathbf{R}^{6d})$,
$\omega_{1},\omega_{2}\in\mathscr{P}_{E}(\mathbf{R}^{2d})$ and
$\vartheta\in\mathscr{P}_{E}(\mathbf{R}^{2d})$ be such that (0.12)′ and (2.10)
hold. If $a\in\mathcal{M}^{p,\infty,p,q,q,q}_{(\omega)}(\mathbf{R}^{3d})$,
then $\operatorname{Op}(a)$ from $\Sigma_{1}(\mathbf{R}^{d})$ to
$\Sigma_{1}^{\prime}(\mathbf{R}^{d})$ is uniquely extendable to a continuous
map from $M^{p_{1}}_{\sigma,(\omega_{1})}(\mathbf{R}^{d})$ to
$M^{p_{2}}_{\sigma,(\omega_{2})}(\mathbf{R}^{d})$, and

$\displaystyle\|\operatorname{Op}(a)f\|_{M^{p_{2}}_{\sigma,(\omega_{2})}}$
$\displaystyle\lesssim\|a\|_{\mathcal{M}^{p,\infty,p,q,q,q}_{(\omega)}}\|f\|_{M^{p_{1}}_{\sigma,(\omega_{1})}},$
(3.5) $\displaystyle a$
$\displaystyle\in\mathcal{M}^{p,\infty,p,q,q,q}_{(\omega)}(\mathbf{R}^{3d}),\
f\in M^{p_{1}}_{\sigma,(\omega_{1})}(\mathbf{R}^{d}).$

As a special case of the previous result we have the following extension of
Theorem 0.1 in the introduction.

######  Theorem 0.1′.

Let $p,q,p_{j},q_{j},r\in(0,\infty]$, $j=1,2$, be such that

$\frac{1}{p_{2}}-\frac{1}{p_{1}}=\frac{1}{q_{2}}-\frac{1}{q_{1}}=\frac{1}{p}+\min\left(0,\frac{1}{q}-1\right),\quad
q\leq p_{2},q_{2}\leq p,$

and (3.4) holds with strict inequality when $q>\min(1,p)$. Also let
$\omega\in\mathscr{P}_{E}(\mathbf{R}^{6d})$,
$\omega_{1},\omega_{2}\in\mathscr{P}_{E}(\mathbf{R}^{2d})$ and
$\vartheta\in\mathscr{P}_{E}(\mathbf{R}^{2d})$ be such that (0.12)′ and (2.10)
hold. If $a\in\mathcal{M}^{p,\infty,p,q,q,q}_{(\omega)}(\mathbf{R}^{3d})$,
then $\operatorname{Op}(a)$ is continuous from
$M^{p_{1},q_{1}}_{(\omega_{1})}(\mathbf{R}^{d})$ to
$M^{p_{2},q_{2}}_{(\omega_{2})}(\mathbf{R}^{d})$, and from
$W^{p_{1},q_{1}}_{(\omega_{1})}(\mathbf{R}^{d})$ to
$W^{p_{2},q_{2}}_{(\omega_{2})}(\mathbf{R}^{d})$.

We observe that if $p=\infty$ and $q\in(0,1]$ in Theorem 0.1′, then
$p_{1}=p_{2}\in[q,\infty]$, $q_{1}=q_{2}\in[q,\infty]$ and we may choose
$r=\infty$ and thereby choose $\vartheta(\eta,z)=1$ everywhere. It is now
evident that Theorem 0.1′ takes the form of Theorem 0.1 in the introduction
for such choices of $p$ and $q$.

By combining Theorem 3.3 or Proposition 3.4 with (1.26) instead of Theorem 1.8
we get the following. The details are left for the reader.

######  Theorem 3.7.

Let $p,q,r\in[1,\infty]$ be such that and

$\frac{1}{q^{\prime}}\leq\frac{1}{r},$ (3.6)

hold. Also let $\omega\in\mathscr{P}_{E}(\mathbf{R}^{6d})$,
$\omega_{1},\omega_{2}\in\mathscr{P}_{E}(\mathbf{R}^{2d})$ and
$\vartheta\in\mathscr{P}_{E}(\mathbf{R}^{2d})$ be such that (0.12)′ and (2.10)
hold. If $a\in\mathcal{W}^{p,\infty,p,q,q,q}_{(\omega)}(\mathbf{R}^{3d})$,
then $\operatorname{Op}(a)$ from $\Sigma_{1}(\mathbf{R}^{d})$ to
$\Sigma_{1}^{\prime}(\mathbf{R}^{d})$ is uniquely extendable to a continuous
map from $M^{q^{\prime},p^{\prime}}_{(\omega_{1})}(\mathbf{R}^{d})$ to
$W^{p,q}_{(\omega_{2})}(\mathbf{R}^{d})$, and

$\displaystyle\|\operatorname{Op}(a)f\|_{W^{p,q}_{(\omega_{2})}}$
$\displaystyle\lesssim\|a\|_{\mathcal{W}^{p,\infty,p,q,q,q}_{(\omega)}}\|f\|_{M^{q^{\prime},p^{\prime}}_{(\omega_{1})}},$
(3.7) $\displaystyle a$
$\displaystyle\in\mathcal{W}^{p,\infty,p,q,q,q}_{(\omega)}(\mathbf{R}^{3d}),\
f\in M^{q^{\prime},p^{\prime}}_{\sigma,(\omega_{1})}(\mathbf{R}^{d}).$

###  3.2. Pseudo-differential operators with amplitudes in weighted Gevrey
analogies of the Hörmander class $S_{0,0}^{0}$

We shall next apply Theorem 0.1 to deduce some continuity properties for
amplitude type pseudo-differential operators with certain smooth symbols when
acting on modulation spaces. For any
$\omega\in\mathscr{P}_{E}(\mathbf{R}^{d})$ and $s\geq 1$, let

$\displaystyle\|f\|_{S^{(\omega)}_{s;h}}$
$\displaystyle\equiv\sup_{\alpha\in{\mathbf{N}}^{d}}\left(\frac{\|(\partial^{\alpha}f)/\omega\|_{L^{\infty}}}{h^{|\alpha|\alpha!^{s}}}\right)$
(3.8) $\displaystyle S^{(\omega)}(\mathbf{R}^{d})$ $\displaystyle=\\{\,f\in
C^{\infty}(\mathbf{R}^{d})\,;\,\|(\partial^{\alpha}f)/\omega\|_{L^{\infty}}<\infty,\
\text{for every}\ \alpha\in{\mathbf{N}}^{d}\,\\}$ (3.9) $\displaystyle
S^{(\omega)}_{s}(\mathbf{R}^{d})$ $\displaystyle=\\{\,f\in
C^{\infty}(\mathbf{R}^{d})\,;\,\|f\|_{S^{(\omega)}_{s;h}}<\infty,\ \text{for
some}\ h>0\,\\}$ (3.10) and $\displaystyle S^{(\omega)}_{0,s}(\mathbf{R}^{d})$
$\displaystyle=\\{\,f\in
C^{\infty}(\mathbf{R}^{d})\,;\,\|f\|_{S^{(\omega)}_{s;h}}<\infty,\ \text{for
every}\ h>0\,\\}$ (3.11)

For $\omega\in\mathscr{P}_{E}(\mathbf{R}^{d})$ and $s\geq 1$, set

$\displaystyle\omega_{r}(x,\xi)=\omega(x)\langle\xi\rangle^{-r}\quad\text{and}\quad\omega_{r,s}(x,\xi)=\omega(x)e^{-r|\xi|^{\frac{1}{s}}}.$
(3.12)

The following lemma is a straight-forward consequence of [1, Proposition 2.5],
[30, Proposition 2.7], Proposition 1.5 (2) and the fact that

$M^{p,\infty}_{(\omega_{2})}(\mathbf{R}^{d})\subseteq
M^{p,q}_{(\omega_{1})}(\mathbf{R}^{d})\quad\text{when}\quad q\in(0,\infty],\
N>\frac{d}{q},\ \omega_{2}(x,\xi)=\omega_{1}(x,\xi)\langle\xi\rangle^{N}$

(see also [8]). The details are left for the reader. Here recall Subsection
1.1 for the definition of involved weight classes.

######  Lemma 3.8.

Let $q\in(0,\infty]$, $s\geq 1$, $\omega\in\mathscr{P}_{E}(\mathbf{R}^{d})$,
and let $\omega_{r}$ and $\omega_{r,s}$ be as in (3.12). Then the following is
true:

  1. (1)

if in addition $\omega\in\mathscr{P}(\mathbf{R}^{d})$, then

$S^{(\omega)}(\mathbf{R}^{d})=\underset{r>0}{\textstyle{\bigcap}}M^{\infty,q}_{(1/\omega_{r})}(\mathbf{R}^{d})\text{;}$

  2. (2)

if in addition $\omega\in\mathscr{P}_{E,s}^{0}(\mathbf{R}^{d})$, then

$S^{(\omega)}_{s}(\mathbf{R}^{d})=\underset{r>0}{\textstyle{\bigcup}}M^{\infty,q}_{(1/\omega_{r,s})}(\mathbf{R}^{d})\text{;}$

  3. (3)

if in addition $\omega\in\mathscr{P}_{E,s}(\mathbf{R}^{d})$, then

$S^{(\omega)}_{0,s}(\mathbf{R}^{d})=\underset{r>0}{\textstyle{\bigcap}}M^{\infty,q}_{(1/\omega_{r,s})}(\mathbf{R}^{d})\text{.}$

We have now the following. Here the involved weights should satisfy

$\frac{\omega_{1}(y,\xi)}{\omega_{2}(x,\xi)}\gtrsim\omega_{0}(x,y,\xi).$
(3.13)

######  Proposition 3.9.

Let $p,q\in(0,\infty]$ and $s\geq 1$. Then the following is true:

  1. (1)

if $\omega_{0}\in\mathscr{P}(\mathbf{R}^{3d})$,
$\omega_{j}\in\mathscr{P}(\mathbf{R}^{2d})$, $j=1,2$, satisfy (3.13) and $a\in
S^{(\omega_{0})}(\mathbf{R}^{3d})$, then $\operatorname{Op}(a)$ is continuous
from $M^{p,q}_{(\omega_{1})}(\mathbf{R}^{d})$ to
$M^{p,q}_{(\omega_{2})}(\mathbf{R}^{d})$;

  2. (2)

if $\omega_{0}\in\mathscr{P}_{E,s}(\mathbf{R}^{3d})$,
$\omega_{j}\in\mathscr{P}_{E,s}(\mathbf{R}^{2d})$, $j=1,2$, satisfy (3.13) and
$a\in S^{(\omega_{0})}_{0,s}(\mathbf{R}^{3d})$, then $\operatorname{Op}(a)$ is
continuous from $M^{p,q}_{(\omega_{1})}(\mathbf{R}^{d})$ to
$M^{p,q}_{(\omega_{2})}(\mathbf{R}^{d})$;

  3. (3)

if $\omega_{0}\in\mathscr{P}_{E,s}^{0}(\mathbf{R}^{3d})$,
$\omega_{j}\in\mathscr{P}_{E,s}^{0}(\mathbf{R}^{2d})$, $j=1,2$, satisfy (3.13)
and $a\in S^{(\omega_{0})}_{s}(\mathbf{R}^{3d})$, then $\operatorname{Op}(a)$
is continuous from $M^{p,q}_{(\omega_{1})}(\mathbf{R}^{d})$ to
$M^{p,q}_{(\omega_{2})}(\mathbf{R}^{d})$.

###### Proof.

We only prove (3). The other assertions follow by similar argument and are
left for the reader.

Since all weights are moderate, (3.13) gives

$\frac{\omega_{2}(x,\xi)}{\omega_{1}(z,\zeta)}\lesssim\omega_{0}(x,z,\zeta)^{-1}e^{r|\xi-\zeta|^{\frac{1}{s}}}$

for every $r>0$. Hence, using that

$e^{r|x+y|^{\frac{1}{s}}}\leq e^{r|x|^{\frac{1}{s}}}e^{r|y|^{\frac{1}{s}}}$

we obtain

$\frac{\omega_{2}(x,\xi)}{\omega_{1}(z,\zeta)}\lesssim\omega_{0}(x,z,\zeta+\eta)^{-1}e^{r(|\xi-\zeta|^{\frac{1}{s}}+|\xi-\zeta-\eta|^{\frac{1}{s}}+|z-x|^{\frac{1}{s}})}$

for every $r>0$. The result now follows by combining Theorem 0.1 with Lemma
3.8. ∎

###  3.3. Schatten and Nuclearity properties for pseudo-differential operators
of amplitude type

Next we shall combine Theorem 3.3 with results in [51] to find Schatten-von
Neumann and nuclear properties for pseudo-differential operators of amplitude
type with symbols in suitable modulation spaces.

Let $p\in(0,\infty]$, $\mathscr{B}_{1}$ and $\mathscr{B}_{2}$ be quasi-Banach
spaces, and let $T\in\mathcal{L}(\mathscr{B}_{1},\mathscr{B}_{2})$. Then the
singular number of $T$ of order $j\geq 1$ is defined by

$\sigma_{j}(T)\equiv\inf\|T-T_{j}\|_{\mathcal{L}(\mathscr{B}_{1},\mathcal{B}_{2})},$

where the infimum is taken over all
$T_{j}\in\mathcal{L}(\mathscr{B}_{1},\mathscr{B}_{2})$ with rank at most
$j-1$. Then $\mathscr{I}_{p}(\mathscr{B}_{1},\mathscr{B}_{2})$ consists of all
$T\in\mathcal{L}(\mathscr{B}_{1},\mathscr{B}_{2})$ such that

$\|T\|_{\mathscr{I}_{p}(\mathscr{B}_{1},\mathscr{B}_{2})}\equiv\|\\{\sigma_{j}(T)\\}_{j=1}^{\infty}\|_{\ell^{p}(\mathbf{Z}_{+})}$

is finite. We observe that
$\mathscr{I}_{\infty}(\mathscr{B}_{1},\mathscr{B}_{2})=\mathcal{L}(\mathscr{B}_{1},\mathscr{B}_{2})$
with the same quasi-norms. We recall that if in addition $\mathscr{B}_{1}$ and
$\mathscr{B}_{2}$ are Hilbert spaces, then
$\mathscr{I}_{2}(\mathscr{B}_{1},\mathscr{B}_{2})$ and
$\mathscr{I}_{1}(\mathscr{B}_{1},\mathscr{B}_{2})$ are the sets of Hilbert-
Schmidt and trace class operators from $\mathscr{B}_{1}$ to $\mathscr{B}_{2}$,
respectively.

The following result is now a straight-forward consequence of [50, Theorem
3.4] and Theorem 3.3. The details are left for the reader.

######  Theorem 0.2′.

Let $p,q,r\in(0,\infty]$ be such that $q\leq\min(p,p^{\prime})$ and (3.4)
hold, $\omega\in\mathscr{P}_{E}(\mathbf{R}^{6d})$ and
$\omega_{1},\omega_{2}\in\mathscr{P}_{E}(\mathbf{R}^{2d})$ be such that
(0.12)′ holds with $\vartheta$ being the same as in Theorem 0.1′. If
$a\in\mathcal{M}^{p,\infty,p,q,q,q}_{(\omega)}(\mathbf{R}^{3d})$, then
$\operatorname{Op}(a)\in\mathscr{I}_{p}(M^{2}_{(\omega_{1})}(\mathbf{R}^{d}),M^{2}_{(\omega_{2})}(\mathbf{R}^{d}))$,
and

$\|\operatorname{Op}(a)\|_{\mathscr{I}_{p}(M^{2}_{(\omega_{1})},M^{2}_{(\omega_{2})})}\lesssim\|a\|_{\mathcal{M}^{p,\infty,p,q,q,q}_{(\omega)}},\quad
a\in\mathcal{M}^{p,\infty,p,q,q,q}_{(\omega)}(\mathbf{R}^{3d}).$

Next we perform similar discussions for nuclear operators. Let $p\in(0,1]$,
$\mathscr{B}_{1}$ be a Banach space with dual $\mathscr{B}_{1}^{\prime}$,
$\mathscr{B}_{2}$ be a quasi-Banach space, and let
$T\in\mathcal{L}(\mathscr{B}_{1},\mathscr{B}_{2})$. Then $T$ is said to be in
the class $\mathscr{N}_{p}(\mathscr{B}_{1},\mathscr{B}_{2})$ of nuclear
operators of order $p$, if there are sequences
$\\{\varepsilon_{j}\\}_{j=1}^{\infty}\subseteq\mathscr{B}_{1}^{\prime}$ and
$\\{e_{j}\\}_{j=1}^{\infty}\subseteq\mathscr{B}_{2}$ such that

$Tf=\sum_{j=1}^{\infty}\langle\varepsilon_{j},f\rangle
e_{j}\quad\text{and}\quad\sum_{j=1}^{\infty}(\|\varepsilon_{j}\|_{\mathscr{B}_{1}^{\prime}}\|e_{j}\|_{\mathscr{B}_{2}})^{p}<\infty,\quad
f\in\mathscr{B}_{1}.$ (3.14)

We also set

$\|T\|_{\mathscr{N}_{p}(\mathscr{B}_{1},\mathscr{B}_{2})}\equiv\inf\left(\sum_{j=1}^{\infty}(\|\varepsilon_{j}\|_{\mathscr{B}_{1}^{\prime}}\|e_{j}\|_{\mathscr{B}_{2}})^{p}\right)^{\frac{1}{p}},$

where the infimum is taken over all
$\\{\varepsilon_{j}\\}_{j=1}^{\infty}\subseteq\mathscr{B}_{1}^{\prime}$ and
$\\{e_{j}\\}_{j=1}^{\infty}\subseteq\mathscr{B}_{2}$ such that (3.14) holds
true. (See e. g. [11, 12, 51].)

We observe that $\mathscr{N}_{p}(\mathscr{B}_{1},\mathscr{B}_{2})$ decreases
with $\mathscr{B}_{1}$, and increases with $p$ and $\mathscr{B}_{2}$. If in
addition $\mathscr{B}_{1}$ and $\mathscr{B}_{2}$ are Hilbert spaces, then the
spectral theorem implies that

$\mathscr{N}_{p}(\mathscr{B}_{1},\mathscr{B}_{2})=\mathscr{I}_{p}(\mathscr{B}_{1},\mathscr{B}_{2})$

with the same quasi-norms. (See e. g. [32].) As a consequence we have

$\mathscr{N}_{p}(M^{\infty}_{(\omega_{1})}(\mathbf{R}^{d}),M^{p}_{(\omega_{2})}(\mathbf{R}^{d}))\subseteq\mathscr{N}_{p}(M^{2}_{(\omega_{1})}(\mathbf{R}^{d}),M^{2}_{(\omega_{2})}(\mathbf{R}^{d}))=\mathscr{I}_{p}(M^{2}_{(\omega_{1})}(\mathbf{R}^{d}),M^{2}_{(\omega_{2})}(\mathbf{R}^{d})).$
(3.15)

The following result is now a consequence of [51, Theorem 4.2] and Theorem
3.3. The details are left for the reader.

######  Theorem 0.3′.

Let $p\in(0,1]$, $\omega\in\mathscr{P}_{E}(\mathbf{R}^{6d})$ and
$\omega_{1},\omega_{2}\in\mathscr{P}_{E}(\mathbf{R}^{2d})$ be such that (0.12)
holds. If $a\in\mathcal{M}^{p,\infty,p,p,p,p}_{(\omega)}(\mathbf{R}^{3d})$,
then
$\operatorname{Op}(a)\in\mathscr{N}_{p}(M^{\infty}_{(\omega_{1})}(\mathbf{R}^{d}),M^{p}_{(\omega_{2})}(\mathbf{R}^{d}))$,
and

$\|\operatorname{Op}(a)\|_{\mathscr{N}_{p}(M^{\infty}_{(\omega_{1})},M^{p}_{(\omega_{2})})}\lesssim\|a\|_{\mathcal{M}^{p,\infty,p,p,p,p}_{(\omega)}},\quad
a\in\mathcal{M}^{p,\infty,p,p,p,p}_{(\omega)}(\mathbf{R}^{3d}).$

Due to (3.15) it is evident that Theorem 0.3′ improves Theorem 0.2′ when
$p\in(0,1]$.

######  Remark 3.10.

There are other nuclearity properties for pseudo-differential operators of the
types $\operatorname{Op}_{A}(a)$ with symbols in modulation spaces, available
in the literature and which are comparable with those in [51] (see e. g. [11,
12]). In this context we have chosen to use the nuclearity results in [51]
because they seems to be sharp.

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