---
abstract: 'We deduce continuity and Schatten-von Neumann properties for operators with matrices satisfying mixed quasi-norm estimates with Lebesgue and Schatten parameters in $(0,\infty ]$. We use these results to deduce continuity and Schatten-von Neumann properties for pseudo-differential operators with symbols in quasi-Banach modulation spaces, or in appropriate H[ö]{}rmander classes.'
address: 'Department of Mathematics, Linn[æ]{}us University, V[ä]{}xj[ö]{}, Sweden'
author:
- Joachim Toft
title: 'Continuity and compactness for pseudo-differential operators with symbols in quasi-Banach spaces or H[ö]{}rmander classes'
---

Introduction {#sec0}
============

The singular values for a linear operator is a non-increasing sequence of non-negative numbers which are strongly linked to questions on continuity and compactness for the operator in the following sense:

-   the operator is continuous, if and only if its singular values are bounded.

-   the operator is compact, if and only if its singular values decay towards zero at infinity. Moreover, fast decays of the singular values permit efficient finite rank approximations.

-   the operator has rank $j\ge 0$, if and only if its singular values of order $j+1$ and higher are zero.

(See [@Ho1; @Gc2] and Section \[sec1\] for definitions.)

In particular, there is a strong connection between the decay of the singular values and finding pseudo-inverses in convenient ways, since such questions are linked to efficient finite rank approximations.

One way to measure the decay of singular values is to consider Schatten-von Neumann classes. More precisely, let $T$ be a linear operator. Then $T$ belongs to ${\mathscr I}_p$, the set of Schatten-von Neumann operators of order $p\in (0,\infty ]$, if and only if its singular values $\sigma _1(T),\sigma _2(T),\dots$ belong to $\ell ^p$. Since the singular values are non-negative and non-increasing, it follows that $$\label{SiVaSchattProp}
\begin{alignedat}{3}
\sigma _j(T) &= o (j^{-1/p}),&
\quad &\text{when} &\quad
T&\in {\mathscr I}_p,\ p<\infty ,
\\[1ex]
\sigma _j(T) &\neq o (j^{-1/p}),&
\quad &\text{when} &\quad
T&\notin {\mathscr I}_{p+{\varepsilon}},\ p<\infty ,\ {\varepsilon}>0,
\end{alignedat}$$ which indicates the link between Schatten-von Neumann classes and the decays of singular values.

It is in general a difficult task to find exact and convenient characterizations of Schatten-von Neumann classes. One is therefore left to find suitable necessary or sufficient conditions when characterizing such classes. For example a Toeplitz operator, acting on $L^2$ belongs to ${\mathscr I}_p$, $p\in [1,\infty]$, when its symbol belongs to $L^p$ (cf. [@HeWo; @BaCoIs; @BaIs]). For pseudo-differential operators ${\operatorname{Op}}(a)$, the situation is slightly different since ${\operatorname{Op}}(a)$ might not be in ${\mathscr I}_p$, $p\neq 2$, when its symbol $a$ belongs to $L^p$. On the other hand, by adding further restrictions on the symbols it is possible to deduce similar sufficient conditions as for Toeplitz operators. For example, if $S(m,g)$ is an appropriate H[ö]{}rmander class parameterized with the Riemannian metric $g$ and weight function $m$ on the phase space, then $$\label{HormSchattenEquiv}
{\{ \, {\operatorname{Op}}_t(a)\, ;\, a\in S(m,g)\, \} }\subseteq {\mathscr I}_p
\qquad \Longleftrightarrow \qquad
m\in L^p.$$ (Cf. Theorems 2.1 and 2.9 in [@BuTo]. See also [@Ho0; @Ho1; @Toft4] for pre-results.)

There are several Schatten-von Neumann results for pseudo-differential operators with symbols in modulation spaces, Besov spaces and Sobolev spaces (cf. [@Toft5] and the references therein). In particular, let $M^{p,q}$ be the classical modulation space with parameters $p,q\in [1,\infty ]$, introduced by Feichtinger in [@Fei1]. Then $$\begin{gathered}
{\operatorname{Op}}(a)  \in {\mathscr I}_p 
\quad  \text{when} \quad
a\in M^{p,p}, \ p\in [1,2],\label{ModSchatten1}
\\[1ex]
{\operatorname{Op}}(a) \, :\, M^{p_1,q_1}\to M^{p_1,q_1} 
\quad  \text{when} \quad
a\in M^{\infty ,1},\ p_1,q_1\in [1,\infty ],
\label{ModCont1}
\intertext{and}
{\operatorname{Op}}(a) \, :\, M^{\infty ,\infty }\to M^{1,1} 
\quad  \text{when} \quad
a\in M^{1 ,1}.
\label{ModCont1B}\end{gathered}$$ The relation was essentially deduced by Gr[ö]{}chenig and Heil, although it seems to be well-known earlier by Feichtinger (cf. [@GH1 Proposition 4.1]). The relation was first proved in [@Gc2], with certain pre-results given already in [@GH1; @Sj1], and is in some sense obtained by Feichtinger already in [@Fei1].

There are also several extensions and modifications of these results. For example, in [@GH2; @Toft2] it was proved that $$\label{ModCont1C}
\begin{gathered}
{\operatorname{Op}}(a)\, :\, M^{p_1,q_1}\to M^{p_2,q_2}
\quad \text{when} \quad
a\in M^{p,q}, \ q\le \min (p,p')
\\[1ex]
\quad \text{and} \quad
\frac 1{p_1} - \frac 1{p_2} = \frac 1{q_1} - \frac 1{q_2} = 1-\frac 1{p} - \frac 1{q},
\ q\le p_2,q_2\le p,
\end{gathered}$$ which covers both and . See also [@Toft3; @Toft5; @Toft8; @Toft11] for extensions of the latter result to weighted spaces, and [@MoPf; @Wa] for related results with other types of modulation spaces as symbol classes. Furthermore, in [@DeRu1; @DeRu2; @DeRu3; @FiRu], related analysis in background of compact or local-compact Lie groups can be found.

In the literature, it is usually assumed that $p$ and $q$ here above belong to $[1,\infty ]$ instead of the larger interval $(0,\infty ]$. An important reason for excluding the cases $p<1$ or $q<1$ is that the involved spaces fail to be local convex, leading in general to several additional difficulties compared to the situation when $p,q\in [1,\infty ]$. On the other hand, in view of it is valuable to decide whether an operator belongs to ${\mathscr I}_p$ or not, also in the case $p<1$. Here we remark that convenient Schatten-$p$ results with $p<1$ can be found for Hankel and Toeplitz operators in e.g. [@Is], and for pseudo-differential operators on compact Lie groups in e.g. [@DeRu1; @DeRu2; @DeRu3].

In the paper we deduce weighted extensions of –, where in contrast to [@Toft3; @Toft5; @Toft8; @Toft11], the case $p<1$ is included. First we deduce continuity and Schatten-von Neumann properties for suitable types of matrix operators. Thereafter we carry over these results to the case of pseudo-differential operators with symbols in modulation spaces, using Gabor analysis as link, in analogous ways as in e.g. [@GH1; @Gc3; @Gc4; @GrSt; @Toft5; @WaSc].

Here we remark that our analysis is comprehensive compared to [@GH1; @Gc3; @Gc4; @GrSt; @Toft5; @WaSc] because of the absent of local-convexity. The situation is handled by using the Gabor analysis in [@GaSa; @Toft12], for non-local convex modulation spaces, in combination of suitable factorization techniques for matrix operators.

In order to shed some more light we explain some consequences of our investigations. As a special case of Theorem \[thmOpSchatten\] we have $$\tag*{(\ref{ModSchatten1})$'$}
{\operatorname{Op}}(a) \in {\mathscr I}_p
\quad \text{when} \quad
a\in M^{p,p}, \ p\in (0,2],$$ i.e. still holds after $[1,2]$ is replaced by the larger interval $(0,2]$. Furthermore, we prove that $'$ is sharp in the sense that any modulation space (with trivial weight), and not contained in $M^{p,p}({\mathbf R^{2d}})$, contains symbols, whose corresponding pseudo-differential operators fail to belong to ${\mathscr I}_p$ (cf. Theorem \[SchattenConverse\]).

In Section \[sec3\] we also deduce general continuity results for pseudo-differential operators with symbols in modulation spaces. In particular, is extended in Theorem \[thmOpCont\] extend in several ways, and as special case, and are extended into $$\begin{gathered}
\tag*{(\ref{ModCont1})$'$}
{\operatorname{Op}}(a) \, :\, M^{p_1,q_1}\to M^{p_1,q_1}
\\[1ex]
\text{when} \quad
a\in M^{\infty ,q},\ p_1,q_1\in [q,\infty ],\ q\in (0,1],\end{gathered}$$ and $$\tag*{(\ref{ModCont1B})$'$}
{\operatorname{Op}}(a) \, :\, M^{\infty ,\infty }\to M^{q,q} 
\quad  \text{when} \quad
a\in M^{q ,q},\ q\in (0,1].$$

In Section \[sec4\] we apply $'$ to deduce Schatten-von Neumann properties for pseudo-differential operators with symbols in $S(m,g)$ in H[ö]{}rmander-Weyl calculus. In particular we show that the sufficiency part of still holds for $p\in (0,1]$ (cf. Theorem \[thm:WeylHorm1\]). That is, for suitable $m$ and $g$, we have $${\{ \, {\operatorname{Op}}_t(a)\, ;\, a\in S(m,g)\, \} }\subseteq {\mathscr I}_p
\quad \text{when} \quad
m\in L^p.$$

An important part behind the analysis concerns Theorem \[factorizationprop\], which in the non-weighted case, essentially state that any matrix $A\in \mathbb U^{p_0}$ can be factorized as $$\label{MatrixFaktor0}
A=A_1\cdot A_2,
\quad \text{when}\quad
A_j\in \mathbb U^{p_j},\quad
\frac 1{p_1}+\frac 1{p_2}= \frac 1{p_0}.$$ From these relations we obtain $$\label{embSimpleCase}
\mathbb U^{p}\subseteq {\mathscr I}_{p},\quad
\text{when}\quad p\in (0,2].$$

In fact, the set of Hilbert-Schmidt operators on $\ell ^2$ agrees with $\mathbb U^2$, and with ${\mathscr I}_2$ (also in norms). Consequently, $\mathbb U^{2}
={\mathscr I}_2$, and H[ö]{}lder’s inequality for Schatten-von Neumann classes together with give that for every $A\in \mathbb U^{2/N}$, with integer $N\ge 1$, there are matrices $A_1,\dots ,A_N\in \mathbb U^{2}$ such that $$A = A_1\cdots A_N \in \mathbb U^{2}\circ \cdots \circ \mathbb U^{2}
= {\mathscr I}_2\circ \cdots \circ {\mathscr I}_2 = {\mathscr I}_{2/N}.$$ Hence $\mathbb U^{2/N}\subseteq {\mathscr I}_{2/N}$ for every integer $N\ge 1$. A (real) interpolation argument between the cases $$\mathbb U^{2/N}\subseteq {\mathscr I}_{2/N}\quad
\text{and}\quad
\mathbb U^{2} =  {\mathscr I}_{2}$$ now shows that that $\mathbb U^{p}\subseteq {\mathscr I}_{p}$ when $p\in [2/N,2]$. Since $2/N$ can be chosen arbitrarily close to $0$, follows.

In Section \[sec2\], the previous arguments are used to deduce more general versions of involving weighted spaces. (See Theorem \[MatrixSchatten\].)

In Section \[sec5\] we show some applications and other results for Schatten-von Neumann symbols. Here we introduce the set $s_{t,p}^q$ consisting of all symbols $a$ such that ${\operatorname{Op}}_t(a)$ belongs to ${\mathscr I}_p$ and such that the orthonormal sequences of the eigenfunctions to $|{\operatorname{Op}}_t(a)|$ and $|{\operatorname{Op}}_t(a)^*|$ are bounded sets in the modulation space $M^{2q}$. It follows that $s_{t,p}^q$ is contained in $s_{t,p}$ the set of all symbols $a$ such that ${\operatorname{Op}}_t(a)\in {\mathscr I}_p$.

We prove that ${\mathscr S}$ is continuously embedded in $s_{t,p}^q$, and that $$s_{t,p}^p \cap {\mathscr E}' \subseteq {\mathscr F}L^p\cap {\mathscr E}' \subseteq s_{t,p} \cap {\mathscr E}'$$ for every $p>0$.

Finally we remark that in [@DeRu1; @DeRu2; @DeRu3; @FiRu], Delgado, Fischer, Ruzhansky and Turunen deal with various kinds of continuity and compactness questions for pseudo-differential operators acting on functions defined on suitable Lie groups. In their approach, matrix-valued symbols appear naturally, and several interesting results on matrices are deduced. A part of these investigations are related to the analysis in Section \[sec2\].

Acknowledgement {#acknowledgement .unnumbered}
===============

The author is very grateful to K. H. Gr[ö]{}chenig, for valuable advices and comments, leading to several improvements of the content and the style. He is also grateful to P. Wahlberg for valuable comments.

Preliminaries {#sec1}
=============

In this section we recall some facts on Gelfand-Shilov spaces, modulation spaces and Schatten-von Neumann classes. The proofs are in general omitted.

Weight functions
----------------

We start by discussing general properties on weight functions. A *weight* on ${\mathbf R^{d}}$ is a positive function $\omega
\in  L^\infty _{loc}({\mathbf R^{d}})$ such that $1/\omega \in  L^\infty _{loc}({\mathbf R^{d}})$. We usually assume that $\omega$ is *moderate*, or *$v$-moderate* for some positive function $v \in
 L^\infty _{loc}({\mathbf R^{d}})$. This means that $$\label{moderate}
\omega (x+y) \lesssim \omega (x)v(y),\qquad x,y\in {\mathbf R^{d}}.$$ Here $A\lesssim B$ means that $A\le cB$ for a suitable constant $c>0$, and for future references, we write $A\asymp B$ when $A\lesssim B$ and $B\lesssim A$. We note that implies that $\omega$ fulfills the estimates $$\label{moderateconseq}
v(-x)^{-1}\lesssim \omega (x)\lesssim v(x),\quad x\in {\mathbf R^{d}}.$$ Furthermore, if $v$ in can be chosen as a polynomial, then $\omega$ is called a weight of *polynomial type*. We let ${\mathscr P}({\mathbf R^{d}})$ and ${\mathscr P}_E({\mathbf R^{d}})$ be the sets of all weights of polynomial type and moderate weights on ${\mathbf R^{d}}$, respectively.

It can be proved that if $\omega \in {\mathscr P}_E({\mathbf R^{d}})$, then $\omega$ is $v$-moderate for some $v(x) = e^{r|x|}$, provided the positive constant $r>0$ is chosen large enough (cf. [@Gc2.5]). In particular, shows that for any $\omega \in {\mathscr P}_E({\mathbf R^{d}})$, there is a constant $r>0$ such that $$\label{WeightExpEst}
e^{-r|x|}\lesssim \omega (x)\lesssim e^{r|x|},\quad x\in {\mathbf R^{d}}$$ (cf. [@Gc2.5]).

We say that $v$ is *submultiplicative* if $v$ is even and holds with $\omega =v$. In the sequel, $v$ and $v_j$ for $j\ge 0$, always stand for submultiplicative weights if nothing else is stated.

Gelfand-Shilov spaces
---------------------

Next we recall the definition of Gelfand-Shilov spaces.

Let $h,s\in \mathbf R_+$ be fixed. Then $\mathcal S_{s,h}({\mathbf R^{d}})$ is the set of all $f\in C^\infty ({\mathbf R^{d}})$ such that $${\Vert f\Vert _{\mathcal S_{s,h}}}\equiv \sup \frac {|x^\beta \partial ^\alpha
f(x)|}{h^{|\alpha + \beta |}(\alpha !\, \beta !)^s}$$ is finite. Here the supremum is taken over all $\alpha ,\beta \in
\mathbf N^d$ and $x\in {\mathbf R^{d}}$.

Obviously $\mathcal S_{s,h}({\mathbf R^{d}})$ is a Banach space which increases as $h$ and $s$ increase, and is contained in ${\mathscr S}({\mathbf R^{d}})$, the set of Schwartz functions on ${\mathbf R^{d}}$. Furthermore, if $s>1/2$, or $s=1/2$ and $h$ is sufficiently large, then is dense in $\mathscr S$. Hence, the dual $(\mathcal S_{s,h})'({\mathbf R^{d}})$ of $\mathcal S_{s,h}({\mathbf R^{d}})$ is a Banach space which contains $\mathscr S'({\mathbf R^{d}})$.

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 $h$. This implies that $$\label{GSspacecond1}
\mathcal S_s({\mathbf R^{d}}) = \bigcup _{h>0}\mathcal S_{s,h}({\mathbf R^{d}})
\quad \text{and}\quad \Sigma _{s}({\mathbf R^{d}}) =\bigcap _{h>0}\mathcal
S_{s,h}({\mathbf R^{d}}),$$ and that the topology for $\mathcal S_s({\mathbf R^{d}})$ is the strongest possible one such that each inclusion map from $\mathcal S_{s,h}({\mathbf R^{d}})$ to $\mathcal S_s({\mathbf R^{d}})$ is continuous. The space $\Sigma _s({\mathbf R^{d}})$ is a Fr[é]{}chet space with semi norms ${\Vert {\, \cdot \, }\Vert _{\mathcal S_{s,h}}}$, $h>0$. Moreover, $\mathcal S _s({\mathbf R^{d}})\neq \{ 0\}$, if and only if $s\ge 1/2$, and $\Sigma _s({\mathbf R^{d}})\neq \{ 0\}$, if and only if $s>1/2$.

For every ${\varepsilon}>0$ and $s>0$, we have $$\Sigma _s ({\mathbf R^{d}})\subseteq \mathcal S_s({\mathbf R^{d}})\subseteq
\Sigma _{s+{\varepsilon}}({\mathbf R^{d}}).$$

The *Gelfand-Shilov distribution spaces* $\mathcal S_s'({\mathbf R^{d}})$ and $\Sigma _s'({\mathbf R^{d}})$ are the projective and inductive limit respectively of $\mathcal S_{s,h}'({\mathbf R^{d}})$. Hence $$\tag*{(\ref{GSspacecond1})$'$}
\mathcal S_s'({\mathbf R^{d}}) = \bigcap _{h>0}\mathcal
S_{s,h}'({\mathbf R^{d}})\quad \text{and}\quad \Sigma _s'({\mathbf R^{d}})
=\bigcup _{h>0} \mathcal S_{s,h}'({\mathbf R^{d}}).$$ By [@Pil], $\mathcal S_s'$ and $\Sigma _s'$ are the duals of $\mathcal S_s$ and $\Sigma _s$, respectively.

The Gelfand-Shilov spaces are invariant or posses convenient mapping properties under several basic transformations. For example they are invariant under translations, dilations, and under (partial) Fourier transformations.

From now on we let $\mathscr F$ be the Fourier transform, given by $$(\mathscr Ff)(\xi )= \widehat f(\xi ) \equiv (2\pi )^{-d/2}\int _{{\mathbf R^{d}}} f(x)e^{-i{\langle x,\xi\rangle} }\, dx$$ 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 $\mathscr S'({\mathbf R^{d}})$, $\mathcal
S_s'({\mathbf R^{d}})$ and $\Sigma _s'({\mathbf R^{d}})$, 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 $L^2({\mathbf R^{d}})$.

Next we recall some mapping properties of Gelfand-Shilov spaces under short-time Fourier transforms. Let $\phi \in \mathscr S({\mathbf R^{d}})$ be fixed. For every $f\in
\mathscr S'({\mathbf R^{d}})$, the *short-time Fourier transform* $V_\phi
f$ is the distribution on ${\mathbf R^{2d}}$ defined by the formula $$\label{defstft}
(V_\phi f)(x,\xi ) =\mathscr F(f\, \overline{\phi ({\, \cdot \, }-x)})(\xi ) =
(f,\phi ({\, \cdot \, }-x)e^{i{\langle {\, \cdot \, },\xi\rangle}}).$$ We recall that if $T(f,\phi )\equiv V_\phi f$ when $f,\phi \in {\mathcal S}_{1/2}({\mathbf R^{d}})$, then $T$ is uniquely extendable to sequentially continuous mappings $$\begin{aligned}
{2}
T\, &:\, & {\mathcal S}_s'({\mathbf R^{d}})\times {\mathcal S}_s({\mathbf R^{d}}) &\to
{\mathcal S}_s '({\mathbf R^{2d}})\bigcap C^\infty ({\mathbf R^{2d}}),
\\[1ex]
T\, &:\, & {\mathcal S}_s'({\mathbf R^{d}})\times {\mathcal S}_s'({\mathbf R^{d}}) &\to
{\mathcal S}_s '({\mathbf R^{2d}}),\end{aligned}$$ and similarly when ${\mathcal S}_s$ and ${\mathcal S}_s'$ are replaced by $\Sigma _s$ and $\Sigma _s'$, respectively, or by ${\mathscr S}$ and ${\mathscr S}'$, respectively (cf. [@CPRT10; @Toft8]). We also note that $V_\phi f$ takes the form $$\tag*{(\ref{defstft})$'$}
V_\phi f(x,\xi ) =(2\pi )^{-d/2}\int _{{\mathbf R^{d}}}f(y)\overline {\phi
(y-x)}e^{-i{\langle y,\xi\rangle}}\, dy$$ when $f\in L^p_{(\omega )}({\mathbf R^{d}})$ for some $\omega \in
{\mathscr P}_E({\mathbf R^{d}})$, $\phi \in \Sigma _1({\mathbf R^{d}})$ and $p\ge 1$. Here $L^p_{(\omega )}({\mathbf R^{d}})$, when $p\in (0,\infty ]$ and $\omega \in {\mathscr P}_E({\mathbf R^{d}})$, is the set of all $f\in L^p_{loc} ({\mathbf R^{d}})$ such that $f\cdot \omega \in L^p({\mathbf R^{d}})$.

Mixed quasi-normed space 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 $$\begin{aligned}
{3}
{\Vert g_1\Vert _{L^q({\mathbf R^{d}})}}&<\infty ,&
\quad &\text{where} &\quad
g_1(\xi ) &\equiv {\Vert F({\, \cdot \, },\xi )\omega ({\, \cdot \, },\xi )\Vert _{L^p({\mathbf R^{d}})}}
\intertext{and}
{\Vert g_2\Vert _{L^p({\mathbf R^{d}})}}&<\infty ,&
\quad &\text{where} &\quad
g_2(x) &\equiv {\Vert F(x,{\, \cdot \, })\omega (x,{\, \cdot \, })\Vert _{L^q({\mathbf R^{d}})}},\end{aligned}$$ respectively.

More generally, let $${{\boldsymbol p}}=(p_1,\dots , p_d)\in (0,\infty ]^d,
\quad {\boldsymbol q}=(q_1,\dots , q_d)\in (0,\infty ]^d,$$ $\operatorname {S}_d$ be the set of permutations on $\{ 1,\dots ,d\}$, ${{\boldsymbol p}}\in
(0,\infty ]^d$, $\omega \in {\mathscr P}_E({\mathbf R^{d}})$, 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 $$\begin{aligned}
g_{0,\omega ,\mu}(x_1,\dots ,x_d)
&\equiv |f (x_{\sigma ^{-1}(1)},\dots ,x_{\sigma ^{-1}(d)})
\omega (x_{\sigma ^{-1}(1)},\dots ,x_{\sigma ^{-1}(d)})|,
\\[1ex]
g_{k,\omega ,\mu}(x_{k+1},\dots ,x_d) &\equiv {\Vert g_{k-1,\omega ,\mu}({\, \cdot \, },
x_{k+1},\dots ,x_d) \Vert _{L^{p_k}(\mu _k)}},
\quad k=1,\dots ,d-1 ,
\intertext{and let }
{\Vert f\Vert _{L^{{{\boldsymbol p}}}_{\sigma ,(\omega )}(\mu)}} &\equiv
{\Vert g_{d-1,\omega ,\mu}\Vert _{L^{p_d}(\mu _d)}}.\end{aligned}$$ The mixed quasi-norm space $L^{{{\boldsymbol p}}}_{\sigma ,(\omega )}(\mu)$ of Lebesgue type is defined as the set of all $\mu$-measurable functions $f$ such that ${\Vert f\Vert _{L^{{{\boldsymbol p}}}_{\sigma ,(\omega )}(\mu )}}<\infty$.

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 $$\label{LambdaDef}
\begin{aligned}
\Lambda &= \Lambda _{[\theta ]} = T_\theta {\mathbf Z^{d}} \equiv
{\{ \, (\theta _1j_1,\dots ,\theta _dj_d)\, ;\, (j_1,\dots ,j_d)\in {\mathbf Z^{d}}\, \} } ,
\\[1ex]
\theta &=(\theta _1,\dots ,\theta _d)\in {\mathbf R_*^{d}},
\end{aligned}$$ and $T_\theta$ denotes the diagonal matrix with diagonal elements $\theta _1,\dots ,\theta _d$. In the former case we set $L^{{{\boldsymbol p}}}_{\sigma ,(\omega )}(\mu)=L^{{{\boldsymbol p}}}_{\sigma ,(\omega )}=
L^{{{\boldsymbol p}}}_{\sigma ,(\omega )}({\mathbf R^{d}})$, and in the latter case we set $L^{{{\boldsymbol p}}}_{\sigma ,(\omega )}(\mu) =
\ell ^{{{\boldsymbol p}}}_{\sigma ,(\omega )}(\Lambda )$.

For conveniency we also set $L^{{{\boldsymbol p}}}_{(\omega )}=L^{{{\boldsymbol p}}}_{\sigma ,(\omega )}$ and $\ell ^{{{\boldsymbol p}}} _{(\omega )}=\ell ^{{{\boldsymbol p}}} _{\sigma ,(\omega )}$ when $\sigma$ is the identity map, and we let $\ell (\Lambda )$ be the set of all (complex-valued) sequences on $\Lambda$ and $\ell _0
(\Lambda )$ be the set of all $f\in \ell (\Lambda )$ such that $f(j)\neq 0$ for at most finite numbers of $j$. Furthermore, if $\omega$ is equal to $1$, then we set $$L^{{{\boldsymbol p}}}_{\sigma } =L^{{{\boldsymbol p}}}_{\sigma ,(\omega )},
\quad
\ell ^{{{\boldsymbol p}}} _{\sigma }=\ell ^{{{\boldsymbol p}}} _{\sigma ,(\omega )},
\quad
L^{{{\boldsymbol p}}} = L^{{{\boldsymbol p}}}_{(\omega )}
\quad \text{and}  \quad
\ell ^{{{\boldsymbol p}}} = \ell ^{{{\boldsymbol p}}} _{(\omega )}.$$

Let ${{\boldsymbol p}}= (p_1,\dots ,p_d)\in [0,\infty ]^d$, ${\boldsymbol q}=(q_1,\dots ,q_d)\in [0 ,\infty ]^d$ and $t\in [-\infty ,\infty ]$. Then we use the conventions $${{\boldsymbol p}}\le {\boldsymbol q}\quad \text{and}\quad {{\boldsymbol p}}\le t \quad \text{when}\quad
p_j\le q_j\ \text{and}\ p_j\le t,$$ respectively, for every $j=1,\dots ,d$, and $${{\boldsymbol p}}= {\boldsymbol q}\quad \text{and}\quad {{\boldsymbol p}}= t \quad \text{when}\quad
p_j= q_j\ \text{and}\ p_j= t,$$ respectively, for every $j=1,\dots ,d$. The relations ${{\boldsymbol p}}< {\boldsymbol q}$ and ${{\boldsymbol p}}< t$ are defined analogously. We also let $${{\boldsymbol p}}\pm {\boldsymbol q}=(p_1\pm q_1,\dots ,p_d\pm q_d)
\quad \text{and}\quad
{{\boldsymbol p}}\pm t=(p_1\pm t,\dots ,p_d\pm t),$$ provided the right-hand sides are well-defined and belongs to $[-\infty ,\infty ]^d$. Moreover, we set $1/0=\infty$, $1/\infty =0$ and $1/{{\boldsymbol p}}=(1/p_1,\dots ,1/p_d)$.

We also let $$\max ({{\boldsymbol p}}) \equiv \max (p_1,\dots ,p_d)
\quad \text{and}\quad
\min ({{\boldsymbol p}}) \equiv \min (p_1,\dots ,p_d),$$ and note that if $\max ({{\boldsymbol p}}) <\infty$, then $\ell _0 (\Lambda )$ is dense in $\ell ^{{\boldsymbol p}}_{\sigma ,(\omega )} (\Lambda )$.

Modulation spaces {#subsec1.2}
-----------------

Next we define modulation spaces. Let $\phi \in {\mathcal S}_{1/2}({\mathbf R^{d}})\setminus 0$. For any $p,q\in (0.\infty ]$ and $\omega \in {\mathscr P}_E({\mathbf R^{2d}})$, the modulation spaces $M^{p,q}_{(\omega )}({\mathbf R^{d}})$ and $W^{p,q}_{(\omega )}({\mathbf R^{d}})$ are the sets of all $f\in {\mathcal S}_{1/2}'({\mathbf R^{d}})$ such that $V_\phi f\in
L^{p,q}_{(\omega )}({\mathbf R^{2d}})$ and $V_\phi f\in
L^{p,q}_{*,(\omega )}({\mathbf R^{2d}})$, respectively. We equip these spaces with the quasi-norms $${\Vert f\Vert _{M^{p,q}_{(\omega )}}}\equiv {\Vert V_\phi f\Vert _{L^{p,q}_{(\omega )}}}
\quad \text{and}\quad
{\Vert f\Vert _{W^{p,q}_{(\omega )}}}\equiv {\Vert V_\phi f\Vert _{L^{p,q}_{*,(\omega )}}},$$ respectively. One of the most common types of modulation spaces concerns $M^{p,q}_{(\omega )}({\mathbf R^{d}})$ with $\omega \in {\mathscr P}({\mathbf R^{2d}})$, and are sometimes called standard modulation spaces. They were introduced by Feichtinger in [@Fei1] for certain choices of $\omega$.

More generally, for any $\sigma \in \operatorname S_{2d}$, ${{\boldsymbol p}}\in (0,\infty ]^{2d}$ and $\omega \in {\mathscr P}_E({\mathbf R^{2d}})$, the modulation space $M^{{\boldsymbol p}}_{\sigma ,(\omega )}({\mathbf R^{d}})$ is the set of all $f\in {\mathcal S}_{1/2}'({\mathbf R^{d}})$ such that $V_\phi f\in
L^{{{\boldsymbol p}}}_{\sigma ,(\omega )}({\mathbf R^{2d}})$, and we equip $M^{{{\boldsymbol p}}}_{\sigma (\omega )}({\mathbf R^{d}})$ with the quasi-norm $$\label{modnorm2}
{\Vert f\Vert _{M^{{{\boldsymbol p}}}_{\sigma , (\omega )} }}\equiv
{\Vert V_\phi f\Vert _{L^{{{\boldsymbol p}}}_{\sigma ,(\omega )}}}.$$

For conveniency we set $M^p_{(\omega )}=M^{p,p}_{(\omega)}$, and if $\omega =1$ everywhere, then set $$M^{{\boldsymbol p}}=M^{{\boldsymbol p}}_{\sigma ,(\omega )},\quad
M^{p,q} = M^{p,q}_{(\omega )},\quad
W^{p,q} = W^{p,q}_{(\omega )}
\quad \text{and}\quad
M^{p} = M^{p}_{(\omega )}.$$

In the following propositions we list some properties for modulation spaces, and refer to [@Fei1; @FG1; @Gc2; @Toft5] for proofs.

\[p1.4A\] Let $\sigma \in \operatorname S_{2d}$ and ${{\boldsymbol p}}\in (0,\infty ]^{2d}$. Then the following is true:

1.  if $\omega \in {\mathscr P}_{E}({\mathbf R^{2d}})$, then $\Sigma _1({\mathbf R^{d}})
    \subseteq M^{{{\boldsymbol p}}}_{\sigma ,(\omega )}({\mathbf R^{d}}) \subseteq \Sigma _1'({\mathbf R^{d}})$;

2.  if $\omega \in {\mathscr P}_{E}({\mathbf R^{2d}})$ satisfies for every $r >0$, then ${\mathcal S}_1({\mathbf R^{d}})\subseteq M^{{{\boldsymbol p}}}_{\sigma ,(\omega )}({\mathbf R^{d}})
    \subseteq {\mathcal S}_1'({\mathbf R^{d}})$;

3.  if $\omega \in {\mathscr P}({\mathbf R^{2d}})$, then $\mathscr S({\mathbf R^{d}})\subseteq M^{{{\boldsymbol p}}}_{\sigma ,(\omega )}({\mathbf R^{d}})
    \subseteq \mathscr S '({\mathbf R^{d}})$.

\[p1.4B\] Let $\sigma \in \operatorname S_{2d}$, $r\in (0,1]$, ${{\boldsymbol p}},{{\boldsymbol p}}_j\in
(0,\infty ] ^{2d}$ and $\omega ,\omega _j,v\in {\mathscr P}_{E}({\mathbf R^{2d}})$, $j=1,2$, be such that $r\le {{\boldsymbol p}}$, ${{\boldsymbol p}}_1\le {{\boldsymbol p}}_2$, $\omega _2\lesssim \omega _1$, and $\omega$ is $v$-moderate. Then the following is true:

1.  if $\phi \in M^r_{(v)}({\mathbf R^{d}})\setminus 0$, then $f\in M^{{{\boldsymbol p}}}_{\sigma ,(\omega )}({\mathbf R^{d}})$, if and only if is finite. In particular, $M^{{{\boldsymbol p}}}_{\sigma ,(\omega )}({\mathbf R^{d}})$ is independent of the choice of $\phi \in M^r_{(v)}({\mathbf R^{d}})\setminus 0$. Moreover, $M^{{{\boldsymbol p}}}_{\sigma ,(\omega )}({\mathbf R^{d}})$ is a quasi-Banach space under the quasi-norm in , and different choices of $\phi$ give rise to equivalent quasi-norms.

    If in addition ${{\boldsymbol p}}\ge 1$, then $M^{{{\boldsymbol p}}}_{\sigma ,(\omega )}
    ({\mathbf R^{d}})$ is a Banach space with norm ;

2.  $M^{{{\boldsymbol p}}_1}_{\sigma ,(\omega _1)}({\mathbf R^{d}})\subseteq
    M^{{{\boldsymbol p}}_2}_{\sigma ,(\omega _2)}({\mathbf R^{d}})$.

Next we discuss Gabor expansions, and start by recalling some notions. It follows from the analysis in Chapters 11–14 in [@Gc2] that the operators in the following definition are well-defined and continuous.

\[DefAnSynGabOps\] Let $\Lambda =\Lambda _{[\theta ]}$ be as in , $\omega ,v\in {\mathscr P}_E({\mathbf R^{2d}})$ be such that $\omega$ is $v$-moderate, and let $\phi ,\psi \in M^1_{(v)}({\mathbf R^{d}})$.

1.  The *analysis operator* $C^{\Lambda}_\phi$ is the operator from $M^\infty _{(\omega )}({\mathbf R^{d}})$ to $\ell ^{\infty}_{(\omega)}(\Lambda )$, given by $$C^\Lambda _\phi f \equiv \{ V_\phi f(j,\iota ) \} _{j,\iota \in \Lambda} \text ;$$

2.  The *synthesis operator* $D^{\Lambda}_\psi$ is the operator from $\ell ^\infty _{(\omega )}(\Lambda )$ to $M^\infty _{(\omega)}({\mathbf R^{d}})$, given by $$D^\Lambda _\psi c \equiv \sum _{j,\iota \in \Lambda} c_{j,\iota}
    e^{i{\langle {\, \cdot \, },\iota \rangle}}\phi ({\, \cdot \, }-j)\text ;$$

3.  The *Gabor frame operator* $S^{\Lambda}_{\phi ,\psi}$ is the operator on $M^\infty _{(\omega )}({\mathbf R^{d}})$, given by $D^\Lambda _\psi \circ
    C^\Lambda _\phi$, i.e. $$S^{\Lambda}_{\phi ,\psi}f \equiv \sum _{j,\iota \in \Lambda} V_\phi f(j,\iota )
    e^{i{\langle {\, \cdot \, },\iota \rangle}}\psi ({\, \cdot \, }-j).$$

We usually assume that $\theta _1= \cdots =\theta _d={\varepsilon}>0$, and then we set $\Lambda _{\varepsilon}= \Lambda _{[\theta ]}$.

The proof of the following result is omitted since the result follows from Theorem 13.1.1 and other results in [@Gc2] (see also Theorem S in [@Gc1]).

\[ThmS\] Let $\Lambda$ be as in , $v\in {\mathscr P}_E({\mathbf R^{2d}})$ be submultiplicative, and $\phi \in
M^1_{(v)}({\mathbf R^{d}})\setminus 0$. Then the following is true:

1.  if $$\label{DualFrames}
    \{ e^{i{\langle {\, \cdot \, },\iota \rangle}}\phi ({\, \cdot \, }-j) \} _{j,\iota \in \Lambda}
    \quad \text{and}\quad
    \{ e^{i{\langle {\, \cdot \, },\iota \rangle}}\psi ({\, \cdot \, }-j) \} _{j,\iota \in \Lambda}$$ are dual frames to each others, then $\psi \in M^1_{(v)}({\mathbf R^{d}})$;

2.  there is a constant ${\varepsilon}_0>0$ such that the frame operator $S_{\phi ,\phi}^\Lambda$ is a homeomorphism on $M^1_{(v)}({\mathbf R^{d}})$ and are dual frames, when $\Lambda ={\varepsilon}{\mathbf Z^{d}}$, ${\varepsilon}\in (0,{\varepsilon}_0]$ and $\psi = (S_{\phi ,\phi}^\Lambda )^{-1}\phi$.

We also recall the following restatement of [@Toft12 Theorem 3.7] (see also Corollaries 12.2.5 and 12.2.6 in [@Gc2] and Theorem 3.7 in [@GaSa]). Here and in what follows we let $\Lambda ^2=\Lambda \times \Lambda$.

\[ConseqThmS\] Let $\Lambda$ be as in , ${{\boldsymbol p}}\in
(0,\infty ]^{2d}$, $r\in (0,1]$ be such that $r\le \min ({{\boldsymbol p}})$, $\sigma
\in \operatorname {S}_{2d}$, and let $\omega ,v\in {\mathscr P}_E({\mathbf R^{2d}})$ be such that $\omega$ is $v$-moderate. Also let $\phi ,\psi \in M^r_{(v)}({\mathbf R^{d}})$ be such that are dual frames to each other. Then the following is true:

1.  The operators $S_{\phi ,\psi} ^\Lambda \equiv D_\psi \circ C_\phi$ and $S_{\psi ,\phi} ^\Lambda \equiv D_\phi \circ C_\psi$ are both the identity map on $M^{{\boldsymbol p}}_{\sigma ,(\omega )}({\mathbf R^{d}})$, and if $f\in M^{{\boldsymbol p}}_{(\omega )}({\mathbf R^{d}})$, then $$\begin{aligned}
    f &= \sum _{j,\iota \in \Lambda} (V_\phi f)(j,\iota )
    e^{i{\langle {\, \cdot \, },\iota \rangle}}\psi ({\, \cdot \, }-j)\notag
    \\[1ex]
    &=
    \sum _{j,\iota \in \Lambda } (V_\psi f)(j,\iota )
    e^{i{\langle {\, \cdot \, },\iota \rangle}}\phi ({\, \cdot \, }-j),\label{GabExpForm}\end{aligned}$$ with unconditional norm-convergence in $M^{{\boldsymbol p}}_{\sigma ,(\omega )}$ when $\max ({{\boldsymbol p}}) <\infty$, and with convergence in $M^\infty _{(\omega)}$ with respect to the weak$^*$ topology otherwise;

2.  if $f\in M^\infty _{(1/v)}({\mathbf R^{d}})$, then $$\displaystyle{{\Vert f\Vert _{M^{{\boldsymbol p}}_{\sigma ,(\omega )}}}
    \asymp
    {\Vert V_\phi f\Vert _{\ell ^{{\boldsymbol p}}_{\sigma ,(\omega )} (\Lambda ^2) }}
    \asymp
    {\Vert V_\psi f\Vert _{\ell ^{{\boldsymbol p}}_{\sigma ,(\omega )} (\Lambda ^2)}} }.$$

Let $v$, $\phi$ and $\Lambda$ be as in Proposition \[ThmS\]. Then $(S_{\phi ,\phi}^\Lambda )^{-1}\phi$ is called the *canonical dual window of $\phi$*, with respect to $\Lambda$. We have $$S_{\phi ,\phi}^\Lambda (e^{i{\langle {\, \cdot \, },\iota \rangle}}f({\, \cdot \, }-j)) =
e^{i{\langle {\, \cdot \, },\iota \rangle}}(S_{\phi ,\phi}^\Lambda f)({\, \cdot \, }-j),$$ when $f\in M^\infty _{(1/v)}({\mathbf R^{d}})$ and $j,\iota \in \Lambda$. The series in are called *Gabor expansions of $f$* with respect to $\phi$ and $\psi$.

\[RemThmS\] There are several ways to achieve dual frames satisfying the required properties in Proposition \[ConseqThmS\]. In fact, let $v,v_{0}\in {\mathscr P}_E({\mathbf R^{2d}})$ be submultiplicative such that $\omega $ is $v$-moderate and $L^1_{(v_0)}({\mathbf R^{2d}})\subseteq
L^r({\mathbf R^{2d}})$. Then Proposition \[ThmS\] guarantees that for some choice of $\phi ,\psi \in M^1_{(v_0v)}({\mathbf R^{d}})\subseteq
M^r_{(v)}({\mathbf R^{d}})$ and lattice $\Lambda$ in , the sets in are dual frames to each others, and that $\psi = (S_{\phi ,\phi}^\Lambda )^{-1}\phi$.

In the sequel we usually assume that $\Lambda =\Lambda _{\varepsilon}$, with ${\varepsilon}>0$ small enough such that the hypotheses in Propositions \[ThmS\] and \[ConseqThmS\] are fulfilled, and that the window functions and their duals belong to $M^r_{(v)}$ for every $r>0$. This is always possible, in view of Remark \[RemThmS\].

Classes of matrices {#subsec1.5}
-------------------

In what follows we let $\Lambda$ be a in , $A$ be the complex matrix $(a(j,k))_{j,k\in \Lambda}$, $p,q\in
(0,\infty ]$, $\omega$ be a map from $\Lambda ^2$ to $\mathbf R_+$, and $$\begin{gathered}
\label{haomegadef}
h_{A,p,\omega }(k) \equiv {\Vert H_{A,\omega}({\, \cdot \, },k)\Vert _{\ell ^p}},
\\[1ex]
\text{where}
\quad
H_{A,\omega}(j,k)=a(j,j-k)\omega (j,j-k).\end{gathered}$$

\[matrixset1\] Let $0<p,q\le \infty$, $\Lambda$ be as in and let $\omega$ be a map from $\Lambda ^2$ to $\mathbf R_+$.

1.  The set $\mathbb U_0(\Lambda )$ consists of matrices $(a(j,k))_{j,k\in \Lambda}$ such that at most finite numbers of $a(j,k)$ are non-zero;

2.  The set $\mathbb U^{p,q}(\omega ,\Lambda )$ consists of all matrices $A=(a(j,k))_{j,k\in \Lambda}$ such that $${\Vert A\Vert _{\mathbb U^{p,q}(\omega ,\Lambda )}} \equiv {\Vert h_{A,p,\omega }\Vert _{\ell ^q(\Lambda )}},$$ is finite, where $h_{A,p,\omega }$ is given by . Furthermore, $\mathbb U^{p,q}_0(\omega ,\Lambda )$ is the completion of $\mathbb U_0(\Lambda )$ under the quasi-norm ${\Vert {\, \cdot \, }\Vert _{\mathbb
    U^{p,q}(\omega ,\Lambda )}}$.

For conveniency we set $\mathbb U^p(\omega ,\Lambda )=\mathbb
U^{p,p}(\omega ,\Lambda )$, and if $\omega =1$ everywhere, then we set $\mathbb U^{p,q}(\Lambda )= \mathbb U^{p,q}(\omega ,
\Lambda )$ and $\mathbb U^{p}(\Lambda )=
\mathbb U^{p}(\omega ,\Lambda )$.

Pseudo-differential operators
-----------------------------

Next we recall some properties in pseudo-differential calculus. Let $s\ge 1/2$, $a\in {\mathcal S}_s 
({\mathbf R^{2d}})$, and $t\in \mathbf R$ be fixed. Then the pseudo-differential operator ${\operatorname{Op}}_t(a)$ is the linear and continuous operator on ${\mathcal S}_s ({\mathbf R^{d}})$, given by $$\label{e0.5}
({\operatorname{Op}}_t(a)f)(x)
=
(2\pi  ) ^{-d}\iint a((1-t)x+ty,\xi )f(y)e^{i{\langle x-y,\xi\rangle} }\,
dyd\xi .$$ For general $a\in {\mathcal S}_s'({\mathbf R^{2d}})$, the pseudo-differential operator ${\operatorname{Op}}_t(a)$ is defined as the continuous operator from ${\mathcal S}_s({\mathbf R^{d}})$ to ${\mathcal S}_s'({\mathbf R^{d}})$ with distribution kernel $$\label{atkernel}
K_{a,t}(x,y)=(2\pi )^{-d/2}({\mathscr F}_2^{-1}a)((1-t)x+ty,x-y).$$ Here ${\mathscr F}_2F$ is the partial Fourier transform of $F(x,y)\in
{\mathcal S}_s'({\mathbf R^{2d}})$ with respect to the $y$ variable. This definition makes sense, since the mappings $$\label{homeoF2tmap}
{\mathscr F}_2\quad \text{and}\quad F(x,y)\mapsto F((1-t)x+ty,y-x)$$ are homeomorphisms on ${\mathcal S}_s'({\mathbf R^{2d}})$. In particular, the map $a\mapsto K_{a,t}$ is a homeomorphism on ${\mathcal S}_s'({\mathbf R^{2d}})$.

The standard (Kohn-Nirenberg) representation, $a(x,D)={\operatorname{Op}}(a)$, and the Weyl quantization ${\operatorname{Op}}^w(a)$ of $a$ are obtained by choosing $t=0$ and $t=1/2$, respectively, in and .

\[BijKernelsOps\] By Fourier’s inversion formula, and the kernel theorem [@LozPerTask Theorem 2.2] for operators from Gelfand-Shilov spaces to their duals, it follows that the map $a\mapsto {\operatorname{Op}}_t(a)$ is bijective from ${\mathcal S}_s'({\mathbf R^{2d}})$ to the set of all linear and continuous operators from ${\mathcal S}_s({\mathbf R^{d}})$ to ${\mathcal S}_s'({\mathbf R^{2d}})$.

By Remark \[BijKernelsOps\], it follows that for every $a_1\in {\mathcal S}_s '({\mathbf R^{2d}})$ and $t_1,t_2\in \mathbf R$, there is a unique $a_2\in {\mathcal S}_s '({\mathbf R^{2d}})$ such that ${\operatorname{Op}}_{t_1}(a_1) = {\operatorname{Op}}_{t_2} (a_2)$. By Section 18.5 in [@Ho1], the relation between $a_1$ and $a_2$ is given by $$\label{calculitransform}
{\operatorname{Op}}_{t_1}(a_1) = {\operatorname{Op}}_{t_2}(a_2)
\quad \Longleftrightarrow \quad
a_2(x,\xi )=e^{i(t_1-t_2){\langle D_x ,D_\xi\rangle}}a_1(x,\xi ).$$

We also recall that ${\operatorname{Op}}_t(a)$ is a rank-one operator, i.e. $$\label{trankone}
{\operatorname{Op}}_t(a)f=(2\pi )^{-d/2}(f,f_2)f_1, \qquad f\in {\mathcal S}_s({\mathbf R^{d}}),$$ for some $f_1,f_2\in {\mathcal S}_s '({\mathbf R^{d}})$, if and only if $a$ is equal to the *$t$-Wigner distribution* $$\label{wignertdef}
W_{f_1,f_2}^{t}(x,\xi ) \equiv {\mathscr F}(f_1(x+t{\, \cdot \, })\overline{f_2(x-(1-t){\, \cdot \, })} )(\xi ),$$ of $f_1$ and $f_2$. If in addition $f_1,f_2\in L^2({\mathbf R^{d}})$, then $W_{f_1,f_2}^{t}$ takes the form $$\label{wignertdef2}
W_{f_1,f_2}^{t}(x,\xi ) = (2\pi )^{-d/2}\int _{{\mathbf R^{d}}} f_1(x+ty)
\overline{f_2(x-(1-t)y)}e^{-{\langle y,\xi\rangle}} \, dy.$$ (Cf. [@BoDoOl1].) Since the Weyl case is of peculiar interests, we also set $W_{f_1,f_2}=W_{f_1,f_2}^{t}$, when $t=1/2$.

Schatten-von Neumann classes
----------------------------

Let ${\mathcal B}(V_1,V_2)$ denote the set of all linear and continuous operators from the quasi-normed space $V_1$ to the quasi-normed space $V_2$, and let ${\Vert {\, \cdot \, }\Vert _{{\mathcal B}(V_1,V_2)}}$ denote corresponding quasi-norm. Let ${\mathscr H}_k$, $k=1,2,3$, be Hilbert spaces and $T\in {\mathcal B}({\mathscr H}_1, {\mathscr H}_2)$. Then the *singular value* of $T$ of order $j\ge 1$ is defined by $$\sigma _j(T) = \sigma _j(T ,{\mathscr H}_1, {\mathscr H}_2)
\equiv \inf {\Vert T-T_0\Vert _{{\mathcal B}({\mathscr H}_1, {\mathscr H}_2))}},$$ where the infimum is taken over all linear operators $T_0$ from ${\mathscr H}_1$ to ${\mathscr H}_2$ of rank at most $j-1$. The set ${\mathscr I}_p({\mathscr H}_1, {\mathscr H}_2)$ of Schatten-von Neumann operators from ${\mathscr H}_1$ to ${\mathscr H}_2$ of order $p\in (0,\infty ]$ is the set of all $T\in {\mathcal B}({\mathscr H}_1, {\mathscr H}_2)$ such that $$\label{SchattenNormBanach}
{\Vert T\Vert _{{\mathscr I}_p({\mathscr H}_1, {\mathscr H}_2)}}
\equiv {\Vert \{Ê\sigma _j(T)Ê\}Ê_{j\ge 1}\Vert _{\ell ^p}}$$ is finite. We observe that ${\mathscr I}_p({\mathscr H}_1, {\mathscr H}_2)$ is contained in the set of compact operators from ${\mathscr H}_1$ to ${\mathscr H}_2$, when $p<\infty$.

We recall that if $p_0,p_1,p_2\in (0,\infty ]$, then $$\label{YoungSchatten}
\begin{gathered}
{\Vert T_2\circ T_1\Vert _{{\mathscr I}_{p_0}({\mathscr H}_1,{\mathscr H}_3)}}
\le
{\Vert T_1\Vert _{{\mathscr I}_{p_1}({\mathscr H}_1,{\mathscr H}_2)}}
{\Vert T_2\Vert _{{\mathscr I}_{p_2}({\mathscr H}_2,{\mathscr H}_3)}}
\quad \text{when}
\\[1ex]
T_1\in {\mathscr I}_{p_1}({\mathscr H}_1,{\mathscr H}_2),
\quad T_2\in {\mathscr I}_{p_2}({\mathscr H}_2,{\mathscr H}_3),\quad 
\frac 1{p_1}+\frac 1{p_2}= \frac 1{p_0},
\end{gathered}$$ and refer to [@Si; @BS] for more facts about Schatten-von Neumann classes.

For convenience we set $${\mathscr I}_p(\omega _1,\omega _2) \equiv
{\mathscr I}_p({\mathscr H}_1, {\mathscr H}_2),$$ when ${\mathscr H}_k=M^2_{(\omega _k)}({\mathbf R^{d}})$, for some $\omega _k\in
{\mathscr P}_E({\mathbf R^{2d}})$, $k=1,2$. Moreover, if $t\in \mathbf R$ and then $s_{t,p}(\omega _1,\omega _2)$ is the set of all $a\in {\mathcal S}_{1/2}'({\mathbf R^{2d}})$ such that ${\operatorname{Op}}_{t}(a)\in {\mathscr I}_p(\omega _1,\omega _2)$, and we set $${\Vert a\Vert _{s_{t,p}(\omega _1,\omega _2)}}\equiv {\Vert {\operatorname{Op}}_{t}(a)\Vert _{{\mathscr I}_p(\omega _1,\omega _2)}}.$$ We also set $s_p^w(\omega _1,\omega _2)=s_{t,p}(\omega _1,\omega _2)$ in the Weyl case, i.e. when $t=1/2$. Moreover, if $\omega _1=\omega _2=1$, then we set $s_{t,p}({\mathbf R^{2d}})=s_{t,p}(\omega _1,\omega _2)$ and $s_p^w({\mathbf R^{2d}})=s_p^w(\omega _1,\omega _2)$.

We recall that $s_{t,p}(\omega _1,\omega _2)$ is a quasi-Banach space under the quasi-norm $a\mapsto
{\Vert a\Vert _{s_{t,p}(\omega _1,\omega _2)}}\equiv {\Vert {\operatorname{Op}}_t(a)\Vert _{{\mathscr I}_{p}
(\omega _1,\omega _2)}}$. Furthermore, if in addition $p\ge 1$, then $s_{t,p}(\omega _1,\omega _2)$ is a Banach space.

By Remark \[BijKernelsOps\] it follows that the map $a\mapsto {\operatorname{Op}}_t(a)$ from $s_{t,p}(\omega _1,\omega _2)$ to ${\mathscr I}_p(\omega _1, \omega _2)$ is bijective and norm preserving.

Symplectic vector spaces and H[ö]{}rmander symbol classes {#subsec1.8}
---------------------------------------------------------

A real vector space $W$ of dimension $2d$ is called *symplectic* if there is a non-degenerate and anti-symmetric bilinear form $\sigma$ (the symplectic form). By choosing symplectic coordinates $e_1,\dots ,e_d,{\varepsilon}_1,\dots ,{\varepsilon}_d$ in $W$, it follows that $$\sigma (X,Y) = {\langle y,\xi\rangle} -{\langle x,\eta\rangle} ,$$ with $$X=(x,\xi )= \sum _{k=1}^d(x_je_j+\xi _j{\varepsilon}_j)\in W,
\quad
Y=(y,\eta )= \sum _{k=1}^d(y_je_j+\eta _j{\varepsilon}_j)\in W,$$ which allows us to identify $W$ with the phase space $T^*V$ for some vector space $V$ of dimension $d$, or by $T^*{\mathbf R^{d}}\simeq {\mathbf R^{2d}}$.

The symplectic Fourier transform ${\mathscr F}_\sigma$ is the linear and continuous map on ${\mathcal S}_{1/2}'(W)$, given by $$({\mathscr F}_\sigma a)(X) = \pi ^{-d}\int _W a(Y)e^{2i\sigma (X,Y)}\, dY$$ when $a\in {\mathscr S}(W)$. If $X=(x,\xi )\in T^*{\mathbf R^{d}}=W$, then it follows that $({\mathscr F}_\sigma a)(X) = 2^d\widehat a(-2\xi ,2x)$.

Next we recall some notions on H[ö]{}rmander symbol classes, $S(m,g)$, parameterized by the Riemannian metric $g$ and the weight function $m$ on the $2d$ dimensional symplectic vector space $W$ (see e.g. [@BoCh; @BuTo; @Ho0; @Ho1; @Le; @Toft4]). The reader who is not interested of the Schatten-von Neumann results in Section \[sec4\] of pseudo-differential operators with symbols in $S(m,g)$ may pass to the next section.

The H[ö]{}rmander class $S(m,g)$ consists of all $a\in C^\infty (W)$ such that $${\Vert a\Vert _{m,N}}^g \equiv \sum _{k=0}^N \sup _{X\in W}(|a|_k^g(X)/m(X)),
\quad \text{where}\quad
|a|_k^g(X) = \sup |a^{(k)}(X;Y_1,\dots ,Y_k)|.$$ Here the latter supremum is taken over all $Y_1,\dots ,Y_k\in W$ such that $g_X(Y_j)\le 1$, $j=1,\dots ,k$, and $|a|_0^g(X)$ is interpreted as $|a(X)|$.

We need to add some conditions on $m$ and $g$. The metric $g$ is called *slowly varying* if there are positive constants $c$ and $C$ such that $$\label{Eq:SlowlyVar}
C^{-1}g_X \le g_Y \le Cg_X,\quad \text{when}\quad  X,Y\in W$$ satisfy $g_X(X-Y)\le c$, and $m$ is called $g$-continuous when holds with $m(X)$ and $m(Y)$ in place of $g_X$ and $g_Y$, respectively, provided $g_X(X-Y)\le c$.

For the Riemannian metric $g$ on $W$, the *dual* metric $g^\sigma$ with respect to the symplectic form $\sigma$, and the *Planck’s function* $h_g$ are defined by $$g_X^\sigma (Z) \equiv \sup _{g_X(Y)\le 1}\sigma (Y,Z)^2
\qquad \text{and}\qquad
h_g(X) \equiv \sup _{g_X^\sigma (Y)\le 1}g_X(Y)^{1/2}.$$ Moreover, if $g$ is slowly varying and $m$ is $g$-continuous, then $g$ is called $\sigma$-temperate if there are positive constants $C$ and $N$ such that $$\label{Eq:Sigmatemp}
g_Y(Z)\le Cg_X(Z) (1+g_Y(X-Y))^N,\qquad X,Y,Z\in W,$$ and $m$ is called $(\sigma ,g)$-temperate if it is $g$-continuous and holds with $m(X)$ and $m(Y)$ in place of $g_X(Z)$ and $g_Y(Z)$, respectively.

Let $g$ be a Riemannian metric on W. Then $g$ is called *feasible* if it is slowly varying and $h_g\le 1$ everywhere. Furthermore, $g$ is called *strongly feasible* if it is feasible and $\sigma$-temperate.

We remark that the H[ö]{}rmander class $S^r_{\rho ,\delta}$ in [@Ho1], the ${\operatorname{SG}}$-class in [@Co; @Pa], the Shubin classes in [@Sh Definition 23.1] and other well-known families of symbol classes are given by $S(m.g)$ for suitable choices of strongly feasible metrics $g$ and $(\sigma ,g)$-temperate weights $m$.

Estimates for matrices {#sec2}
======================

In this section we deduce continuity and Schatten-properties for matrices in the classes $\mathbb U^{p ,q}(\omega ,\Lambda )$. In the first part we achieve convenient factorization results for matrices in the case when $p=q$ (cf. Theorem \[factorizationprop\]). Thereafter we establish the continuity properties (cf. Theorem \[matrixcont2\]). In the last part of the section we combine these factorizations and continuity results to establish Schatten properties for matrix operators (cf. Theorem \[MatrixSchatten\]).

Theorem \[factorizationprop\] below allows factorizations of matrices in $\mathbb U^{p}(\omega ,\Lambda )$ in suitable ways, when $\Lambda$ is given by . Here the involved weights should fulfill $$\begin{aligned}
{2}
\omega _1(j,j)\omega _2(j,k)&\le \omega _0(j,k),&
\qquad j,k &\in \Lambda \label{weightcond1}
\intertext{or}
\omega _1(j,k)\omega _2(k,k)&\le \omega _0(j,k),&
\qquad j,k &\in \Lambda \label{weightcond2}\end{aligned}$$ and the involved Lebesgue exponents should satisfy the H[ö]{}lder condition $$\label{holdercond}
\frac 1{p_0} \le \frac 1{p_1} + \frac 1{p_2},$$

\[factorizationprop\] Let $\Lambda$ be as in , $p _l\in (0, \infty ]$ be such that hold, $\omega _l$, $l=0,1,2$, be weights on ${\mathbf R^{2d}}$ , and let $A_0\in
\mathbb U^{p_0}(\omega _0,\Lambda )$. Then the following is true:

1.  if holds, then $A_0=A_1\cdot A_2$ for some $A_l\in \mathbb U^{p_l}(\omega _l,\Lambda )$, $l=1,2$. Furthermore, $A_1$ can be chosen as a diagonal matrix;

2.  if holds, then $A_0=A_1\cdot A_2$ for some $A_l\in \mathbb U^{p_l}(\omega _l,\Lambda )$, $l=1,2$. Furthermore, $A_2$ can be chosen as a diagonal matrix.

Moreover, the matrices in [[(1)]{}]{} and [[(2)]{}]{} can be chosen such that $$\label{multcont}
{\Vert A_1\Vert _{\mathbb U^{p_1}(\omega _1,\Lambda )}}
{\Vert A_2\Vert _{\mathbb U^{p_2}(\omega _2,\Lambda )}} \le {\Vert A_0\Vert _{\mathbb U^{p_0}(\omega _0,\Lambda )}}.$$

It is no restrictions to assume that equality is attained in , and by transposition it also suffices to prove (1).

We only prove the result for $p_0<\infty$. The small modifications to the case when $p_0=\infty$ are left for the reader. Let $a(j,k)$ be the matrix elements for $A_0$, and let $A_1 =(b(j,k))$ and $A_2=(c(j,k))$ be the matrices such that $$b(j,k) =
\begin{cases}
\big (\omega _1(j,j)\big )^{-1}
\displaystyle{\left (\sum _{m}|a(j,m)\omega _0(j,m)|^{p_0}\right
)^{1/{p_1}}},&\quad j=k
\\[3ex]
0,&\quad   j\neq k
\end{cases}$$ and $c(j,k)=a(j,k)/b(j,j)$ when $b(j,j)\neq 0$, and $c(j,k)=0$ otherwise.

Since $$b(j,j)\ge (\omega _1(j,j))^{-1}|a(j,k)\omega _0(j,k)|^{p_0/p_1},
\quad \text{and}\quad
\frac 1{p_0} - \frac 1{p_1} = \frac 1{p_2},$$ gives $$\begin{gathered}
|c(j,k)\omega _2(j,k)|\le |a(j,k)|^{p_0/p_2}\omega _1(j,j)\omega _2(j,k)
/\omega _0(j,k)^{p_0/p_1}
\\[1ex]
\le |a(j,k)|^{p_0/p_2}\omega _0(j,k)^{p_0/p_2}.\end{gathered}$$ This in turn gives $$\begin{gathered}
{\Vert A_1\Vert _{\mathbb U^{p_1}(\omega _1,\Lambda )}} = \left ( \sum _{j,k}|b(j,k)
\omega _1(j,k)|^{p_1}\right )^{1/{p_1}}
\\[1ex]
=
\left (   \left (  \sum _j  \left (   \sum _{m}   |a(j,m)\omega _0(j,m)|^{p_0}
\right )^{1/p_1}  \right )^{p_1}  \right )^{1/p_1} =
{\Vert A_0\Vert _{\mathbb U^{p_0}(\omega _0,\Lambda )}}^{p_0/p_1},\end{gathered}$$ and $$\begin{gathered}
{\Vert A_2\Vert _{\mathbb U^{p_2}(\omega _2,\Lambda )}} =
\left (\sum _{j,k} |c(j,k)\omega _2(j,k)| ^{p_2}\right )^{1/{p_2}}
\\[1ex]
\le 
\left ( \sum _{j,k} |a(j,k)\omega _0(j,k)|^{p_0}\right )^{1/{p_2}} =
{\Vert A_0\Vert _{\mathbb U^{p_0}(\omega _0,\Lambda )}}^{p_0/p_2}.\end{gathered}$$ Hence $A_l\in \mathbb U^{p_l}(\omega _l,\Lambda )$, $l=1,2$. Since $A_0=A_1\cdot
A_2$ and $p_0/p_1+p_0/p_2=1$, the result follows.

If the weights $\omega _l$, $l=0,1,2$, fulfill $$\label{weightcond3}
\omega _1(j,m)\omega _2(m,k)\le \omega _0(j,k),\qquad
\text{for every}\ j,k,m\in \Lambda ,$$ then it is evident that both and are fulfilled. Hence the following result is a special case of Theorem \[factorizationprop\].

\[factorizationprop2\] Let $\Lambda$ be as in , $p _l\in (0, \infty ]$ and let $\omega _l$, $l=0,1,2$, be weights on ${\mathbf R^{2d}}$ such that and hold, and let $A_0\in \mathbb U^{p_0}(\omega _0,\Lambda )$. Then $A_0=A_1\cdot A_2$ for some $A_l\in \mathbb U^{p_l}(\omega _l,\Lambda )$, $l=1,2$. Moreover, the matrices $A_1$ and $A_2$ can be chosen such that holds.

Next we deduce continuity results for matrix operators. We recall that if $A=(a(j,k))_{j,k\in \Lambda}$ is a matrix, then $Af$ is uniquely defined as an element in $\ell (\Lambda )$ when $f\in \ell _0(\Lambda )$, i.e. $$\begin{aligned}
{2}
A\, &:\, &  \ell _0(\Lambda ) &\mapsto \ell (\Lambda ). \label{Abmap1}
\intertext{Furthermore, if in addition $A$ belongs to $\mathbb U_0(\Lambda )$,
then $Af$ is uniquely defined as an element in $\ell _0(\Lambda )$ when $f\in l(\Lambda )$,
i.{\,}e.}
A\, &:\, & \ell (\Lambda ) &\mapsto \ell _0(\Lambda )\phantom ,\qquad \text{when}\quad
A\in \mathbb U_0(\Lambda ). \label{Abmap2}\end{aligned}$$

For $p\in [1,\infty ]$, its conjugate exponent $p'\in [1,\infty ]$ is usually defined by $1/p+1/p'=1$. For $p$ belonging to the larger interval $(0,\infty ]$ it is convenient to extend the definition of $p'$ as $$p'=
\begin{cases}
\ \ 1, & p=\infty
\\[1ex]
\displaystyle{\frac p{p-1}}, & 1<p<\infty
\\[1ex]
\ \ \infty , & 0<p\le 1 .
\end{cases}$$

The next theorem is the main result concerning the continuity for matrix operators.

\[matrixcont2\] Let $\sigma \in S_d$, $\theta \in \mathbf R^d_*$, $\Lambda =T_\theta {\mathbf Z^{d}}$, $\omega _l$ be weights on $\Lambda$, $l=1,2$, and $\omega _0$ be a weight on $\Lambda \times \Lambda$ such that holds. Also let ${{\boldsymbol p}}_1,{{\boldsymbol p}}_2\in (0,\infty]^n$, and $p,q\in (0,\infty]$ be such that $$\label{pqconditions}
\frac 1{{{\boldsymbol p}}_2}-\frac 1{{{\boldsymbol p}}_1} = \frac 1{p}+\min \left ( 0,\frac 1{q}-1\right ) ,
\quad q \le \min ({{\boldsymbol p}}_2)\le
\max ({{\boldsymbol p}}_2) \le p,
$$ and let $A\in \mathbb U^{p,q}(\omega _0,\Lambda )$. Then $A$ from $\ell _0(\Lambda )$ to $\ell (\Lambda )$ is uniquely extendable to a continuous map from $\ell ^{{{\boldsymbol p}}_1} _{\sigma ,(\omega _1)}(\Lambda )$ to $\ell ^{{{\boldsymbol p}}_2}
_{\sigma ,(\omega _2)}(\Lambda )$, and $$\label{Anormest}
{\Vert A\Vert _{{\mathcal B}(\ell ^{{{\boldsymbol p}}_1}_{\sigma ,(\omega _1)} (\Lambda )
, \ell ^{{{\boldsymbol p}}_2}_{\sigma ,(\omega _2)}(\Lambda ))}}\le
{\Vert A\Vert _{\mathbb U^{p,q}(\omega _0,\Lambda )}}.$$

We note that is the same as $$\label{Anormest2}
{\Vert Af\Vert _{\ell ^{{{\boldsymbol p}}_2}_{\sigma ,(\omega _2)}(\Lambda )}}\le {\Vert A\Vert _{\mathbb U^{p,q}(\omega _0,\Lambda )}}
{\Vert f\Vert _{\ell ^{{{\boldsymbol p}}_1}_{\sigma ,(\omega _1)} (\Lambda )}},\quad
f\in \ell ^{{{\boldsymbol p}}_1}_{\sigma ,(\omega _1)} (\Lambda ).$$

By permutation of the Lebesgue exponents, we reduce ourself to the case when $\sigma$ is the identity map. We consider the cases $q\le 1$ and $q\ge 1$ separately. Let $f\in \ell ^{{{\boldsymbol p}}_1}_{(\omega _1)}$, $h=h_{A,\infty ,\omega}$ be the same as in Definition \[matrixset1\], with $\omega =\omega _0$, and set $$c(k) = |f(k)\omega _1(k)|,
\quad
a_0 (j,k) = |a(j,j-k)\omega _0(j,j-k)|
\quad \text{and}\quad
g = Af.$$

First we consider the case when $p=\infty$, $q\le 1$, and in addition $A\in \mathbb U_0(\Lambda )$. Then ${{\boldsymbol p}}_1={{\boldsymbol p}}_2$, and we get $$\begin{gathered}
\label{MatrixComp1}
|g(j)\omega _2(j)|\le \sum _k  |a(j,k)\omega _0(j,k)| \, c(k) 
\\[1ex]
= \sum _k  a_0(j,k) \, c(j-k) 
\\[1ex]
\le \sum _k h(k) c(j-k) =(h*c)(j).\end{gathered}$$ Hence, Corollary 2.2 in [@Toft12] gives $$\label{MatrixComp2}
{\Vert Af\Vert _{\ell ^{{{\boldsymbol p}}_2}_{(\omega _2)}}} = {\Vert g\cdot \omega _2\Vert _{\ell ^{{{\boldsymbol p}}_2}}}\le {\Vert h\Vert _{\ell ^q}}{\Vert c\Vert _{\ell ^{{{\boldsymbol p}}_1}}}
= {\Vert A\Vert _{\mathbb U^{\infty ,q}(\omega _0,\Lambda )}}
{\Vert f\Vert _{\ell ^{{{\boldsymbol p}}_1}_{(\omega _1)}}},$$ and the result follows in this case.

For general $A\in \mathbb U^{\infty ,q}(\omega _0,\Lambda )$ we decompose $A$ and $f$ into $$A=A_1-A_2 +i(A_3-A_4)
\quad \text{and}\quad 
f=f_1-f_2 +i(f_3-f_4),$$ where $A_j$ and $f_k$ only have non-negative entries, chosen as small as possible. By Beppo Levi’s theorem and the estimates above it follows that $A_jf_k$ is uniquely defined as an element in $\ell ^{{{\boldsymbol p}}_1}_{(\omega _1)}$. It also follows from these estimates holds, and we have proved the result in the case $p=\infty$ and $q\le 1$.

The case when $q\le 1$, $p<\infty$ and $A\in \mathbb U_0(\Lambda )$ is obtained by induction. Let $$\begin{aligned}
{2}
G_0(j) &\equiv |g(j)\omega _2(j)|, &
\quad
b_0(j,k)&\equiv a_0(j,k) = |a(j,j-k)\omega _0(j,j-k)|,
\\[1ex]
c_0(j) &\equiv c(j), & \quad
\Lambda _0 &\equiv \{ 0\}
\quad \text{and}\quad
\Lambda _0^* \equiv \Lambda = \theta _1\mathbf Z\times \cdots \times \theta _d\mathbf Z. \end{aligned}$$ Also let $$\begin{aligned}
{2}
{{\boldsymbol p}}_{l,m} &\equiv  (p_{l,1},\dots ,p_{l,m}) &
\quad \text{when}\quad
{{\boldsymbol p}}_{l} &= (p_{l,1},\dots ,p_{l,d}),\ l=1,2,
\\[1ex]
\Lambda _m &\equiv \theta _1\mathbf Z \times \cdots \times \theta _m\mathbf Z, &
\quad
\Lambda _m^* &\equiv \theta _{m+1}\mathbf Z \times \cdots \times \theta _d\mathbf Z,
\\[1ex]
{\boldsymbol j}_m &= (j_{m+1},\dots ,j_d)\in \Lambda _m^* &
\quad \text{and}\quad
{\boldsymbol k}_m &= (k_{m+1},\dots ,k_d)\in \Lambda _m^*
\intertext{when}
j&=(j_1,\dots ,j_d)\in \Lambda &
\quad \text{and}\quad
k&=(k_1,\dots ,k_d)\in \Lambda ,\end{aligned}$$ and let $$b_m({\boldsymbol j}_m,{\boldsymbol k}_m) \equiv {\Vert a_{0,m}({\boldsymbol j}_m,{\, \cdot \, },{\boldsymbol k}_m)\Vert _{\ell ^q)(\Lambda _m)}}, \quad
m =1,\dots ,d,$$ where $$a_{0,m}({\boldsymbol j}_m,k) \equiv {\Vert a_0({\, \cdot \, },{\boldsymbol j}_m,k)\Vert _{\ell ^p(\Lambda _m)}}.$$ Define inductively $$\begin{gathered}
G_m({\boldsymbol j}_m) \equiv {\Vert G_{m-1}({\, \cdot \, },{\boldsymbol j}_m)\Vert _{\ell ^{p_{2,m}} (\theta _m\mathbf Z)}}
\quad \text{and}\quad
c_m({\boldsymbol j}_m) \equiv {\Vert c_{m-1}({\, \cdot \, },{\boldsymbol j}_m)\Vert _{\ell ^{p_{1,m}} (\theta _m\mathbf Z)}},\end{gathered}$$ when $m=1,\dots ,d$, where $\Lambda _d^*$, $G_d$ and $c_d$ are interpreted as $$\{ 0\} ,\quad
{\Vert G_{d-1}\Vert _{\ell ^{p_{2,d}}(\theta _d\mathbf Z)}} = {\Vert Af\Vert _{\ell ^{{{\boldsymbol p}}_2}_{(\omega _2)}}}
\quad \text{and} \quad
{\Vert c_{d-1}\Vert _{\ell ^{p_{1,d}}(\theta _d\mathbf Z)}} = {\Vert f\Vert _{\ell ^{{{\boldsymbol p}}_1}_{(\omega _1)}}},$$ respectively. We claim $$\label{Eq:GbcIneq}
G_m({\boldsymbol j}_m) \le
\left (
\sum _{{\boldsymbol k}_m\in \Lambda _m^*}\big ( b_m({\boldsymbol j}_m,{\boldsymbol k}_m)
c_m({\boldsymbol j}_m -{\boldsymbol k}_m)\big )^q,
\right )^{1/q}$$ for $m=0,\dots ,d$.

In fact, the case $m=0$ follows from the equality in and the fact that $q\le 1$. Suppose is true for $m-1$ in place of $m$, and let $r=p_{2,m}/q$. Then $r\in [1,\infty )$, since $p<\infty$ and $q\le p_{2,m}$. Hence, , and H[ö]{}lder’s and Minkowski’s inequalities in combination with the inductive assumptions give $$\begin{gathered}
G_m({\boldsymbol j}_m)^q
\le
\left (
\sum _{j_m\in \theta _m\mathbf Z}
\left (
\sum _{{\boldsymbol k}_{m-1}\in \Lambda _{m-1}^*}(b_{m-1}({\boldsymbol j}_{m-1},{\boldsymbol k}_{m-1})
c_{m-1}({\boldsymbol j}_{m-1}-{\boldsymbol k}_{m-1})
)^q
\right )^r
\right )^{1/r}
\\[1ex]
\le
\sum _{{\boldsymbol k}_{m-1}\in \Lambda _{m-1}^*}
\left (
\sum _{j_m\in \theta _m\mathbf Z} 
(b_{m-1}({\boldsymbol j}_{m-1},{\boldsymbol k}_{m-1})
c_{m-1}({\boldsymbol j}_{m-1}-{\boldsymbol k}_{m-1})
)^{p_{2,m}}
\right )^{1/r}
\\[1ex]
\le
\sum _{{\boldsymbol k}_{m-1}\in \Lambda _{m-1}^*}
{\Vert b_{m-1}({\, \cdot \, },{\boldsymbol j}_m,{\boldsymbol k}_{m-1})\Vert _{\ell ^p(\theta _m\mathbf Z)}}^q
{\Vert c_{m-1}({\, \cdot \, },{\boldsymbol j}_{m}-{\boldsymbol k}_m)\Vert _{\ell ^{p_{1,m}}(\theta _m\mathbf Z)}}^q.\end{gathered}$$

We have $c_m({\boldsymbol j}_m)= {\Vert c_{m-1}({\, \cdot \, },{\boldsymbol j}_m)\Vert _{\ell ^{p_{1,m}}(\theta _m\mathbf Z)}}$, and $$\sum _{k_m\in \theta _m\mathbf Z}{\Vert b_{m-1}({\, \cdot \, },{\boldsymbol j}_m,k_m,{\boldsymbol k}_m)\Vert _{\ell ^p(\theta _m\mathbf Z)}}^q
\le
{\Vert a_{0,m}({\boldsymbol j}_m{\, \cdot \, },{\boldsymbol k}_m)\Vert _{\ell ^p(\theta _m\mathbf Z)}}^q,$$ by Minkowski’s inequality, and a combination of the previous inequalities give . Hence, by induction we have that holds for every $m=0,\dots ,d$, and by letting $m=d$ we obtain when $A\in \mathbb U_0(\Lambda )$. The result now follows for general $\mathbb U^{p,q}(\omega _0,\Lambda )$ when $p<\infty$ and $q\le 1$ by the fact that $\mathbb U_0(\Lambda )$ is dense in $\mathbb U^{p,q}(\omega _0,\Lambda )$.

Next we consider the case $q\in (1,\infty ]$, and assume first that $p=\infty$. Then $$\frac 1{{{\boldsymbol p}}_1}+\frac 1q = 1+\frac 1{{{\boldsymbol p}}_2}.$$ Hence, if $A\in \mathbb U^{\infty ,q}(\omega _0,\Lambda )$ and $f\in \ell _0(\Lambda)$, then and Young’s inequality give $${\Vert g\Vert _{\ell ^{{{\boldsymbol p}}_2}_{(\omega _2)}}} \le {\Vert h\Vert _{\ell ^q}}{\Vert c\Vert _{\ell ^{{{\boldsymbol p}}_1}}},$$ and follows in this case as well. Since $\max ({{\boldsymbol p}}_1)<\infty$ when $q>1$, the result follows for general $f\in \ell ^{{{\boldsymbol p}}_1}_{(\omega _1)}(\Lambda)$ from the fact that $\ell _0(\Lambda)$ is dense in $\ell ^{{{\boldsymbol p}}_1}_{(\omega _1)}(\Lambda)$.

For general $p$, the result now follows by multi-linear interpolation between the cases $(p,q)=(1,1)$ and $(p,q)=\{ \infty \} \times [1,\infty ]$, using Theorems 4.4.1 and 5.6.3 in [@BeLo]. The proof is complete.

The following consequence of the previous result is particularly important.

\[matrixcont1\] Let $\Lambda$ be as in , $p\in (0,\infty ]$, $\omega _l$, $l=1,2$ be weights on ${\mathbf R^{d}}$ and $\omega _0$ be a weight on ${\mathbf R^{2d}}$ such that $$\label{weightineq1}
\frac {\omega _2(j)}{\omega _1(k)} \le \omega _0(j,k),\qquad j,k\in \Lambda$$ holds. Also let $A\in \mathbb U^p (\omega _0,\Lambda )$. Then $A$ in is uniquely extendable to a continuous map from $\ell ^{p'} _{(\omega _1)}(\Lambda )$ to $\ell ^p_{(\omega _2)}(\Lambda )$, and $$\label{normest}
{\Vert A\Vert _{{\mathcal B}(\ell ^{p'} _{(\omega _1)}(\Lambda ), \ell ^p_{(\omega _2)}(\Lambda ))}}
\le {\Vert A\Vert _{\mathbb U^p(\omega _0,\Lambda )}}.$$

The next result deals with Schatten-von Neumann properties for matrix operators.

\[MatrixSchatten\] Let $\Lambda$ be as in , $\omega _l$, $l=1,2$ be weights on ${\mathbf R^{d}}$ and $\omega _0$ be a weight on ${\mathbf R^{2d}}$ such that holds. Also let $p\in (0,2]$, and let $A\in \mathbb U^p (\omega _0,\Lambda )$. Then $A\in {\mathscr I}_p(\ell ^2_{(\omega _1)}(\Lambda ),
\ell ^2_{(\omega _2)}(\Lambda ))$, and $$\label{Schattnormest1}
{\Vert A\Vert _{{\mathscr I}_p(\ell ^2_{(\omega _1)}(\Lambda ),\ell ^2_{(\omega _2)}(\Lambda ))}}
\le {\Vert A\Vert _{\mathbb U^p(\omega _0,\Lambda )}}.$$

We may assume that equality is attained in , and that ${\Vert A\Vert _{\mathbb U^p(\omega _0,\Lambda )}}=1$. Then it follows that $${\mathscr I}_2(\ell ^2_{(\omega _1)}(\Lambda ),\ell ^2_{(\omega _2)}(\Lambda )) =
\mathbb U^2 (\omega _0,\Lambda ),$$ with equality in norms.

First assume that $p=2/N$ for some integer $N\ge 3$, and let $A\in \mathbb U^{2/N} (\omega _0,\Lambda )$. Also let $\vartheta _1(j,k)=\omega _2(j)$, $\vartheta _m(j,k)=1$, $j=2,\dots ,N-1$ and $\vartheta _N(j,k)=\omega _1(k)$. By Theorem \[factorizationprop\] we have $$A=A_1\circ \cdots \circ A_N$$ for some $A_m \in \mathbb U^2 (\vartheta _m,\Lambda )$ which satisfy ${\Vert A_m\Vert _{\mathbb U^2 (\vartheta _m,\Lambda )}}\le 1$, $m=1,\dots ,N$.

By we get $$\begin{gathered}
{\Vert A\Vert _{{\mathscr I}_{2/N}(\ell ^2_{(\omega _1)},\ell ^2_{(\omega _2)})}} \le
{\Vert A_1\Vert _{{\mathscr I}_{2}(\ell ^2,\ell ^2_{(\omega _2)})}}
{\Vert A_N\Vert _{{\mathscr I}_{2}(\ell ^2_{(\omega _1)},\ell ^2)}}
\prod _{m=2}^{N-1} {\Vert A_m\Vert _{{\mathscr I}_{2}(\ell ^2,\ell ^2)}}
\\[1ex]
=
\prod _{m=2}^{N-1} {\Vert A_m\Vert _{\mathbb U^2 (\vartheta _m,\Lambda )}} \le 1,\end{gathered}$$ and the result follows in the case $p=2/N$.

The result is therefore true when $p=2/N$ for some integer $N\ge 3$, and when $p=2$. For $p\in [2/N,2]$, the result now follows by (real) interpolation between the cases $p=2$ and $p=2/N$, letting $q$, $p_\theta$, $p_k$, $q_k$ and $\theta \in [0,1]$, $k=0,1$, in Teorema 3.2, (3.11) and (3.13) in [@BS] be chosen such that $$q=p_\theta ,\quad q_0=p_0=\frac 2N ,\quad q_1=p_1=2
\quad \text{and}\quad
\frac 1{p_\theta} = \frac {1-\theta}{p_0}+\frac \theta{p_1}.$$

For general $p\in (0,2]$, the result now follows by choosing $N\ge 3$ such that $p>2/N$. The proof is complete.

Continuity and Schatten-von Neumann properties for pseudo-differential operators {#sec3}
================================================================================

In this section we deduce continuity and Schatten-von Neumann results for pseudo-differential operators with symbols in modulation spaces. In particular we extend results in [@GH1; @Gc2; @Toft2; @Toft5; @Toft11] to include Schatten and Lebesgue parameters less than one.

We start with the following result on continuity.

\[thmOpCont\] Let $t\in \mathbf R$, $\sigma \in \operatorname{S}_{2d}$, $\omega _1,\omega _2
\in \mathscr P_{E}({\mathbf R^{2d}})$ and $\omega _0\in \mathscr P_{E}({\mathbf R^{2d}}\oplus {\mathbf R^{2d}})$ be such that $$\frac {\omega _2(x,\xi  )}{\omega _1
(y,\eta )} \lesssim \omega _0( (1-t)x+ty,t\xi +(1-t)\eta ,\xi -\eta ,y-x ).$$ Also let ${{\boldsymbol p}}_1,{{\boldsymbol p}}_2\in (0,\infty]^{2d}$, $p,q\in (0,\infty]$ be such that hold, and let $a\in M^{p,q}_{(\omega _0)}({\mathbf R^{2d}})$. Then ${\operatorname{Op}}_t(a)$ from $\mathcal S_{1/2}({\mathbf R^{d}})$ to $\mathcal S_{1/2}'({\mathbf R^{d}})$ extends uniquely to a continuous map from $M^{{{\boldsymbol p}}_1}
_{\sigma ,(\omega _1)}({\mathbf R^{d}})$ to $M^{{{\boldsymbol p}}_2}_{\sigma ,(\omega _2)}({\mathbf R^{d}})$, and $$\label{PsDOEst}
{\Vert {\operatorname{Op}}_t(a)\Vert _{{\mathcal B}(M^{{{\boldsymbol p}}_1}_{\sigma ,(\omega _1)},
M^{{{\boldsymbol p}}_2}_{\sigma ,(\omega _2)})}}
\lesssim
{\Vert a\Vert _{M^{p,q}_{(\omega _0)}}}.$$

We need some preparing lemmata for the proof. We recall that $\Lambda ^2=\Lambda \times \Lambda$ when $\Lambda$ is a lattice.

\[aFrames\] Let $v\in {\mathscr P}_E({\mathbf R^{4d}})$, $\phi _1,\phi _2\in \Sigma _1({\mathbf R^{d}})\setminus 0$, and let $$\Phi (x,\xi )=\phi _1(x)\overline {\widehat \phi _2(\xi )}e^{-i{\langle x,\xi\rangle} },$$ Then there is a lattice $\Lambda$ in such that $$\begin{aligned}
&\{ \Phi (x-j,\xi -\iota )e^{i({\langle x,\kappa\rangle} +{\langle k,\xi\rangle} )} \} _{(j, \iota ),(k, \kappa )
\in \Lambda ^2}
\intertext{is a Gabor frame with canonical dual frame}
&\{ \Psi (x-j,\xi -\iota )e^{i({\langle x,\kappa\rangle} +{\langle k,\xi\rangle} )} \}
_{(j,\iota ),(k,\kappa ) \in \Lambda ^2},\end{aligned}$$ where $\Psi = (S_{\Phi ,\Phi }^{\Lambda ^2\times \Lambda ^2})^{-1}\Phi$ belongs to $M^r _{(v)}({\mathbf R^{2d}})$ for every $r>0$.

Note that $\Phi$ in Lemma \[aFrames\] is the Rihaczek (cross)-distribution of $\phi _1$ and $\phi _2$ (cf. [@GrSt]).

The result follows from Remark \[RemThmS\], and the fact that $\Phi \in \Sigma _1({\mathbf R^{2d}})\setminus 0$ in view of [@CaTo Theorem 3.1] or [@CaWa Proposition 3.4] .

\[PsDoDisc\] Let $\Lambda$, $\phi _1$, $\phi _2$, $\Phi$ and $\Psi$ be as in Lemma \[aFrames\]. Also let $v\in {\mathscr P}_E ({\mathbf R^{4d}})$, $a\in M^\infty _{(1/v)}({\mathbf R^{2d}})$, $$\begin{gathered}
c_0({\boldsymbol j},{\boldsymbol k}) \equiv (V_\Psi a)(j,\kappa ,\iota -\kappa ,k-j)e^{i{\langle k-j,\kappa\rangle}},
\\[1ex]
\text{where}\quad {\boldsymbol j}=(j,\iota )\in \Lambda ^2 ,\ {\boldsymbol k}= (k,\kappa )\in \Lambda ^2.\end{gathered}$$ and let $A$ be the matrix $A=(c_0({\boldsymbol j},{\boldsymbol k})_{{\boldsymbol j},{\boldsymbol k}\in \Lambda ^2}$. Then the following is true:

1.  if $p,q\in (0,\infty ]$ and $\omega ,\omega _0\in {\mathscr P}_E({\mathbf R^{4d}})$ satisfy $$\label{omega0omegaRel}
    \omega (x,\xi ,y,\eta )\asymp \omega _0(x,\eta ,\xi -\eta ,y-x),$$ then $a\in M^{p,q}_{(\omega _0)}({\mathbf R^{2d}})$, if and only if $A\in \mathbb U^{p,q}(\omega ,\Lambda ^2 )$, and then $${\Vert a\Vert _{M^{p,q}_{(\omega _0)}}}\asymp
    {\Vert A\Vert _{\mathbb U^{p,q}(\omega ,\Lambda ^2 )}}\text ;$$

2.  ${\operatorname{Op}}(a)$ as a map from ${\mathcal S}_{1/2}({\mathbf R^{d}})$ to ${\mathcal S}_{1/2}'({\mathbf R^{d}})$, given by $$\label{OpaFactorization}
    {\operatorname{Op}}(a) = D_{\phi _1} \circ A \circ C_{\phi _2}.$$

Some arguments in [@GrSt] appear in the proof of Lemma \[PsDoDisc\].

We have $$|c_0({\boldsymbol j},{\boldsymbol j}-{\boldsymbol k})| = | (V_\Psi a) (j,\iota -\kappa ,\kappa ,-k)|.$$ Hence, Proposition \[ConseqThmS\] (2) gives $${\Vert A\Vert _{\mathbb U^{p,q}(\omega ,\Lambda ^2 )}}
=
{\Vert V_\Psi a\Vert _{\ell ^{p,q}_{(\omega _0)}(\Lambda ^2 \times \Lambda ^2 )}}
\asymp
{\Vert a\Vert _{M^{p,q}_{(\omega _0)}}},$$ and (1) follows.

Next we prove (2). Let $f\in {\mathcal S}_{1/2}({\mathbf R^{d}})$, and let $$c({\boldsymbol j},{\boldsymbol k}) = (V_\Psi a)(j,\iota ,\kappa ,k).$$ By Proposition \[ConseqThmS\] we have $$a= \sum _{{\boldsymbol j},{\boldsymbol k}\in \Lambda ^2 }c({\boldsymbol j}, {\boldsymbol k})\Phi _{{\boldsymbol j}, {\boldsymbol k}},$$ where $$\Phi _{{\boldsymbol j}, {\boldsymbol k}}(x,\xi )= e^{i({\langle x,\kappa\rangle} +{\langle k,\xi\rangle})}\Phi (x-j,\xi -\iota ).$$ This gives $${\operatorname{Op}}(a)  = \sum _{{\boldsymbol j},{\boldsymbol k}\in \Lambda ^2 }c({\boldsymbol j}, {\boldsymbol k})
{\operatorname{Op}}(\Phi _{{\boldsymbol j}, {\boldsymbol k}}),$$ and we shall evaluate ${\operatorname{Op}}(\Phi _{{\boldsymbol j}, {\boldsymbol k}})f$.

We have $$\Phi _{{\boldsymbol j}, {\boldsymbol k}}(x,\xi ) = \phi _1(x-j)\overline
{\widehat \phi _2(\xi -\iota )}e^{-i{\langle x-j,\xi -\iota \rangle}}e^{i({\langle x,\kappa\rangle} +{\langle k,\xi\rangle} )},$$ and by straight-forward computations we get $$({\operatorname{Op}}(\Phi _{{\boldsymbol j}, {\boldsymbol k}})f)(x) = \phi _1(x-j)e^{i{\langle x,\iota +\kappa \rangle}}e^{-i{\langle j,\iota\rangle}}
F_0({\boldsymbol j}, {\boldsymbol k}),$$ where $$F_0({\boldsymbol j}, {\boldsymbol k})
=
(2\pi )^{-d/2}\int \widehat f(\xi ) \overline {\widehat \phi _2(\xi -\iota )}
e^{i{\langle j+k,\xi\rangle}}\, d\xi = (V_{\widehat \phi _2}\widehat f)(\iota , -(j+k)).$$ Since $$(V_{\widehat \phi _2}\widehat f)(\xi ,-x) = e^{i{\langle x,\xi\rangle} }V_{\phi _2} f(x,\xi ),$$ we get $$({\operatorname{Op}}(\Phi _{{\boldsymbol j}, {\boldsymbol k}})f)(x) = \big (e^{i{\langle k,\iota\rangle}} V_{\phi _2} f(j+k,\iota )\big )
\phi _1(x-j)e^{i{\langle x,\iota +\kappa \rangle}}.$$

This gives $$\begin{gathered}
\label{OpaComp}
({\operatorname{Op}}(a)f)(x) = \sum _{{\boldsymbol j},{\boldsymbol k}\in \Lambda ^2} (V_\Psi a)(j,\iota ,\kappa ,k)
e^{i{\langle k,\iota\rangle}} V_{\phi _2} f(j+k,\iota )\phi _1(x-j)e^{i{\langle x,\iota +\kappa \rangle}}
\\[1ex]
= \sum _{{\boldsymbol j},{\boldsymbol k}\in \Lambda ^2} (V_\Psi a)(j,\kappa ,\iota -\kappa ,k-j)
e^{i{\langle k-j,\kappa\rangle}} V_{\phi _2} f({\boldsymbol k})\phi _1(x-j)e^{i{\langle x,\iota\rangle}}
\\[1ex]
= \sum _{{\boldsymbol j}\in \Lambda ^2} h({\boldsymbol j})\phi _1(x-j)e^{i{\langle x,\iota\rangle}},\end{gathered}$$ where $$h({\boldsymbol j}) = \sum _{{\boldsymbol k}\in \Lambda ^2} c_0({\boldsymbol j},{\boldsymbol k})
V_{\phi _2} f({\boldsymbol k}).$$ The result now follows from the facts that $h=A\cdot (C_{\phi _2}f)$ and that the right-hand side of is equal to $(D_{\phi _1}h)(x)$.

By Proposition 1.7 in [@Toft12] and its proof, it suffices to prove the result for $t=0$. Let $\omega$, $\omega _0$, $\Lambda$, $\phi _1$, $\phi _2$ and $A$ be as in Lemma \[PsDoDisc\]. Then $$\label{SynthOps}
C_{\phi _2}\, :\, M^{{{\boldsymbol p}}_1}_{(\omega _1)}({\mathbf R^{d}}) \to
\ell ^{{{\boldsymbol p}}_1}_{(\omega _1)}(\Lambda ^2)
\quad \text{and}\quad
D_{\phi _1}\, :\, \ell ^{{{\boldsymbol p}}_2}_{(\omega _2)}(\Lambda ^2 )
\to M^{{{\boldsymbol p}}_2}_{(\omega _2)}({\mathbf R^{d}})$$ are continuous.

Furthermore, since $$\frac {\omega _2(x,\xi )}{\omega _1(y,\eta )}\lesssim
\omega _0(x,\eta ,\xi -\eta ,y-x),$$ it follows from that $$\frac {\omega _2(X)}{\omega _1(Y)}\lesssim
\omega (X ,Y),\quad X=(x,\xi )\in {\mathbf R^{2d}},\ Y=(y,\eta )\in {\mathbf R^{2d}}.$$ holds. Hence Theorem \[matrixcont2\] shows that $$A \, :\, \ell ^{{{\boldsymbol p}}_1}_{(\omega _1)}(\Lambda ^2 ) \to
\ell ^{{{\boldsymbol p}}_2}_{(\omega _2)}(\Lambda ^2 )$$ is continuous. Hence, if ${\operatorname{Op}}(a)$ is defined by , it follows that ${\operatorname{Op}}(a)$ from ${\mathcal S}_{1/2}({\mathbf R^{d}})$ to ${\mathcal S}_{1/2}'({\mathbf R^{d}})$ extends to a continuous map from $M^{{{\boldsymbol p}}_1}_{(\omega _1)}({\mathbf R^{d}})$ to $M^{{{\boldsymbol p}}_2}_{(\omega _2)}({\mathbf R^{d}})$.

It remains to prove the uniqueness of the extension, If $\max ({{\boldsymbol p}}_1 )<\infty$, then the uniqueness follows from the fact that ${\mathcal S}_{1/2}({\mathbf R^{d}})$ is dense in $M^{{{\boldsymbol p}}_1}_{(\omega _1)}({\mathbf R^{d}})$. If instead $p<\infty$, then $q<\infty$, and ${\mathcal S}_{1/2}({\mathbf R^{2d}})$ is dense in $M^{p,q}_{(\omega _0)}({\mathbf R^{2d}})$. The uniqueness now follows in this case from $'$, and the fact that ${\operatorname{Op}}(a)$ is uniquely defined as an operator from ${\mathcal S}_{1/2}'({\mathbf R^{d}})$ to ${\mathcal S}_{1/2}({\mathbf R^{d}})$, when $a\in {\mathcal S}_{1/2}({\mathbf R^{2d}})$.

Finally assume that $p=\infty$ and $\max ({{\boldsymbol p}}_1 ) =\infty$. Then $'$ shows that $q\le 1$. In particular, if $f\in M^{{{\boldsymbol p}}_1}_{(\omega _1)}({\mathbf R^{d}})$ then $f\in M^\infty _{(\omega _1)}({\mathbf R^{d}})$. The uniqueness now follows from the fact that ${\operatorname{Op}}(a)f$ is uniquely defined as an element in $M^\infty _{(\omega _2)}({\mathbf R^{d}})$, in view of Theorem A.2 in [@Toft11].

We have also the following result on Schatten-von Neumann properties for pseudo-differential operators.

\[thmOpSchatten\] Let $t\in \mathbf R$, $\omega _1,\omega _2\in \mathscr P_{E}({\mathbf R^{2d}})$ and $\omega _0\in \mathscr P_{E}({\mathbf R^{2d}}\oplus {\mathbf R^{2d}})$ be such that $$\label{Eq:omegajCond}
\frac {\omega _2(x,\xi  )}{\omega _1
(y,\eta )} \asymp \omega _0( (1-t)x+ty,t\xi +(1-t)\eta ,\xi -\eta ,y-x )$$ Also let $p,p_j,q,q_j\in (0,\infty ]$ be such that $$p_1\le p,\quad q_1\le \min (p,p'),\quad p_2\ge \max (p,1),
\quad q_2\ge \max (p,p').$$ Then $$\label{ModSchattenEmb}
M^{p_1,q_1}_{(\omega _0)}({\mathbf R^{2d}})
\subseteq
s_{t,p}(\omega _1,\omega _2)
\subseteq
M^{p_2,q_2}_{(\omega _0)}({\mathbf R^{2d}})$$ and $$\label{PsDOEstSchatt}
{\Vert a\Vert _{M^{p_2,q_2}_{(\omega _0)}}}\lesssim
{\Vert a\Vert _{s_{t,p}(\omega _1,\omega _2)}}\lesssim
{\Vert a\Vert _{M^{p_1,q_1}_{(\omega _0)}}}.$$

We use the same notations as in the proof of Theorem \[thmOpCont\]. The result is true for $p\in [1,\infty ]$ in view of Theorem A.3 in [@Toft11] and Proposition \[p1.4A\]. Hence it suffices to prove the assertion for $p\in (0,1)$.

By Proposition 1.7 in [@Toft12] and its proof, it suffices to prove the result for $t=0$.

It follows from and that $$\begin{gathered}
{\Vert {\operatorname{Op}}(a)\Vert _{\mathscr I_p(\omega _1, \omega _2)}}
=
{\Vert D_{\phi _1} \circ A \circ C_{\phi _1}\Vert _{{\mathscr I}_p(\omega _1, \omega _2)}}
\\[1ex]
\lesssim
{\Vert D_{\phi _1}\Vert _{{\mathscr I}_\infty {(\ell ^2_{(\omega _2)}(\Lambda ^2 ),M^2_{(\omega _2)})}}}
{\Vert A\Vert _{{\mathscr I}_p {(\ell ^2_{(\omega _1)}(\Lambda ^2 ),
\ell ^2_{(\omega _2)}(\Lambda ^2 ) ) }}}
{\Vert C_{\phi _2}\Vert _{{\mathscr I}_\infty {(M^2_{(\omega _1)},\ell ^2_{(\omega _1)} (\Lambda ^2 )  )}}}
\\[1ex]
\asymp 
{\Vert A\Vert _{{\mathscr I}_p {(\ell ^2_{(\omega _1)}(\Lambda ^2 ),
\ell ^2_{(\omega _2)}(\Lambda ^2 ) ) }}}
\lesssim {\Vert A\Vert _{\mathbb U^{p,p}(\omega ,\Lambda ^2  )}}
\asymp {\Vert a\Vert _{M^{p,p}_{(\omega _0)}}},\end{gathered}$$ and the result follows.

Theorems \[thmOpCont\] and \[thmOpSchatten\] are related to certain results [@MoPf; @Pf] when the involved weights are trivial, and the involved Lebesgue exponents belong the subset $[1,\infty ]$ of $(0,\infty ]$. More precisely, let ${{\boldsymbol p}}\in [1,\infty ]^{4d}$ be given by $${{\boldsymbol p}}=(p_1,\dots ,p_1,p_2,\dots ,p_2,q_1,\dots ,q_1,q_2,\dots ,q_2),$$ and each $p_j$ and $q_j$ occur $d$ times. Then S. Molahajloo and G. E. Pfander investigate in [@MoPf; @Pf], continuity of pseudo-differential operators with symbols in $M^{{{\boldsymbol p}}}({\mathbf R^{2d}})$, when acting between $M^{r_1,s_1}({\mathbf R^{d}})$ and $M^{r_2,s_2}({\mathbf R^{d}})$, for some $r_j,s_j\in [1,\infty ]$ (cf. Theorem 1.3 in [@MoPf]).

We note that there are some overlaps between Theorems \[thmOpCont\] and \[thmOpSchatten\] and the results in [@MoPf; @Pf]. On the other hand, the results in [@MoPf; @Pf], and Theorems \[thmOpCont\] and \[thmOpSchatten\] do not contain each others, since the assumptions on the symbols are more restrictive in Theorems \[thmOpCont\] and \[thmOpSchatten\], while the assumptions on domains and image spaces are more restrictive in [@MoPf; @Pf].

Next we show that Theorem \[thmOpSchatten\] is optimal with respect to $p$. More precisely, we have the following result

\[SchattenConverse\] Let $t\in \mathbf R$, $\omega _k\in \mathscr P_{E}({\mathbf R^{2d}})$, $k=1,2$, and $\omega _0\in \mathscr P_{E}({\mathbf R^{2d}}\oplus {\mathbf R^{2d}})$ be such that holds. Also let $p,q,r\in (0,\infty ]$, and suppose $$\label{SchattenModIncl}
M^{p,q}_{(\omega _0)}({\mathbf R^{2d}})\subseteq s_{t,r}
(\omega _1, \omega _2).$$ Then the following is true:

1.  $p\le r$ and $q\le \min (2,r)$;

2.  if in addition $\omega _1,\omega _2 \in {\mathscr P}({\mathbf R^{2d}})$ and $r\ge 2$, then $q\le p'$.

We need some preparations for the proof. The following result concerning Wigner distributions extends [@Toft11 Proposition A.4] to involve Lebesgue exponents smaller than one (cf. ). We omit the proof since the arguments are the same as in the proof of [@Toft11 Proposition A.4]. (See also [@Gc2; @Toft5] and the references therein for related results.)

\[t-WignerMod\] Let $t\in \mathbf R$, and let $p_j,q_j,p,q\in (0,\infty ]$ be such that $p\le p_j,q_j\le q$, for $j=1,2$, and $$\label{(A.8)}
1/p_1+1/p_2=1/q_1+1/q_2=1/p+1/q.$$ Also let $\omega _1,\omega _2\in {\mathscr P}_E({\mathbf R^{2d}})$ and $\omega \in {\mathscr P}_E({\mathbf R^{2d}}\oplus {\mathbf R^{2d}})$ be such that $$\label{(A.9)}
\omega _0( (1-t)x+ty,t\xi +(1-t)\eta ,\xi -\eta ,y-x )
\lesssim
\omega _1(x,\xi )\omega _2(y,\eta ).$$ Then the map $(f_1,f_2)\mapsto W_{f_1,f_2}^t$ from ${\mathcal S}_{1/2}'({\mathbf R^{d}})\times {\mathcal S}_{1/2}'({\mathbf R^{d}})$ to ${\mathcal S}_{1/2}'({\mathbf R^{2d}})$ restricts to a continuous mapping from $M^{p_1,q_1}_{(\omega _1)}({\mathbf R^{d}})\times
M^{p_2,q_2}_{(\omega _2)}({\mathbf R^{d}})$ to $M^{p,q}_{(\omega _0
)}({\mathbf R^{2d}})$, and $$\label{(A.10)}
{\Vert W_{f_1,f_2}^t\Vert _{M^{p,q}_{(\omega _0)}}}
\lesssim
{\Vert f_1\Vert _{M^{p_1,q_1}_{(\omega _1)}}} {\Vert f_2\Vert _{M^{p_2,q_2}_{(\omega _2)}}}$$ when $f_1,f_2\in {\mathcal S}_{1/2}'({\mathbf R^{d}})$.

We have now the following extension of Corollary 4.2 (1) in [@Toft2].

\[cor4.2\] Let $p\in (0,\infty ]$, $q\in (2,\infty ]$, $t\in \mathbf R$, and let $\omega _2\in {\mathscr P}_E({\mathbf R^{2d}})$ and $\omega _0\in {\mathscr P}_E({\mathbf R^{4d}})$ be such that $$\omega _0((1-t)x,t\xi ,\xi ,-x)\lesssim \omega _2(x,\xi ).$$ Then there is an element $a$ in $M^{p,q}_{(\omega _0)}({\mathbf R^{2d}})$ such that ${\operatorname{Op}}_t(a)$ is not continuous from ${\mathcal S}_{1/2}({\mathbf R^{d}})$ to $M^{2,2}_{(\omega _2)}({\mathbf R^{d}})$.

Let $a=W ^t_{f_2,f_1}$, where $f_1\in \Sigma _1({\mathbf R^{d}})\setminus 0$ and $f_2\in
M^{q,p}_{(\omega _2)}({\mathbf R^{d}})\setminus M^{2,2}_{(\omega _2)}({\mathbf R^{2d}})$. Such choices of $f_2$ are possible in view of Proposition \[ConseqThmS\].

By using the fact that $\omega _0$ and $\omega _2$ are moderate weights, it follows that holds when $\omega _1(x,\xi )
=e^{c(|x|+|\xi |)}$, and the constant $c>0$ is chosen large enough. By Proposition \[p1.4A\], it follows that $f_1\in
M^{p,q}_{(\omega _1)}({\mathbf R^{d}})$. Hence $a\in M^{p,q}_{(\omega _0)}({\mathbf R^{2d}})$ in view of Proposition \[t-WignerMod\].

On the other hand, if $f\in {\mathcal S}_{1/2}({\mathbf R^{d}})\setminus 0$ is chosen such that $f$ and $f_1$ are not orthogonal, then $${\operatorname{Op}}_t(a)f = (f,f_1)\cdot f_2\in
M^{q,p}_{(\omega _2)}({\mathbf R^{d}})\setminus M^{2,2}_{(\omega _2)}({\mathbf R^{2d}}),$$ and the result follows.

We also need the following lemma. We omit the proof since the result is a special case of Proposition 4.3 in [@Toft12]. Here $\check f$ is defined as $\check f(x) = f(-x)$ when $f\in {\mathcal S}_{1/2}'({\mathbf R^{d}})$, and recall from Subsection \[subsec1.8\] that $({\mathscr F}_\sigma a)(X) = 2^d\widehat a(-2\xi ,2x)$ when $X=(x,\xi )
\in {\mathbf R^{2d}}$.

\[identification1\] Let $\omega _1,\omega _2\in {\mathscr P}_E({\mathbf R^{2d}})$, $a\in
\mathscr S'({\mathbf R^{2d}})$, and that $p\in (0,\infty ]$. Then $${\mathscr F}_\sigma (s_p^w(\omega _1,\omega _2))
=
s_p^w(\omega _1,\check \omega _2).$$

We may assume that $t=1/2$, and consider first the case when $1\le r$. Let ${\EuScript M}^{p,q}_{(\omega )}({\mathbf R^{2d}})$ and ${\EuScript W}^{p,q}_{(\omega )}({\mathbf R^{2d}})$ be the modulation spaces when the symplectic Fourier transform is used instead of the ordinary Fourier transform in the definition of modulation space norms of $M^{p,q}_{(\omega )}({\mathbf R^{2d}})$ and $W^{p,q}_{(\omega)}({\mathbf R^{2d}})$, respectively. Then is equivalent to $${\operatorname{Op}}^w({\EuScript M}^{p,q}_{(\omega )}) \subseteq {\mathscr I}_r
(\omega _1,\omega _2),
\quad \text{when}\quad 
\frac {\omega _2(X-Y)}{\omega _1(X+Y)} \asymp \omega (X,Y)$$ (see e.g. [@Toft11]). Let $\phi \in \Sigma ({\mathbf R^{2d}})$ and $\Lambda$ in be chosen such that $\{ \phi (  {\, \cdot \, }-{\boldsymbol j})e^{-2i\sigma ({\, \cdot \, },{\boldsymbol k})}\}
_{{\boldsymbol j},{\boldsymbol k}\in \Lambda ^2}$ is a Gabor frame. Also let $\vartheta ({\boldsymbol k})=\omega (0,{\boldsymbol k})$, $c\in \ell ^\infty _{(\vartheta )}(\Lambda ^2)$, $c_0(0,{\boldsymbol k})
=c({\boldsymbol k})$ and $c_0({\boldsymbol j},{\boldsymbol k}) = 0$ when ${\boldsymbol j}\neq 0$, and let $$a(X) \equiv \sum _{{\boldsymbol k}\in \Lambda ^2}c({\boldsymbol k})\phi (X)
e^{-2i\sigma (X,{\boldsymbol k})}
=
\sum _{{\boldsymbol j},{\boldsymbol k}\in \Lambda ^2}c_0({\boldsymbol j},{\boldsymbol k})
\phi (X-{\boldsymbol j})e^{-2i\sigma (X,{\boldsymbol k})}.$$ Then $a\in {\EuScript M}^{p,\infty}_{(\omega )}({\mathbf R^{2d}})$ for every $p\in (0,\infty ]$. Furthermore, $$\label{aDiffModEq}
a\in {\EuScript M}^{p,q}_{(\omega )}
\quad \Longleftrightarrow \quad
a\in {\EuScript W}^{p,q}_{(\omega )}
\quad \Longleftrightarrow \quad
c\in \ell ^q_{(\vartheta )}$$ holds for every $p,q\in (0,\infty ]$.

Now if $q>r$, then choose $c\in \ell ^q_{(\vartheta )}\setminus \ell
^r_{(\vartheta )}$, and it follows from and that $a\in {\EuScript M}^{p,q}_{(\omega )}
\setminus s^w_r(\omega _1,\omega _2)$. This shows that $q\le r$ when holds.

Assume instead that $p>r$, let $q\in (0,\infty ]$ be arbitrary, choose $c\in \ell ^p_{(\vartheta )}\setminus \ell ^r_{(\vartheta )}$, and consider $$b=\mathscr F_\sigma a \in \mathscr F_\sigma {\EuScript W}^{q,p}_{(\omega )}
= {\EuScript M}^{p,q}_{(\omega _{T})},\qquad \omega _{T}(X,Y)= \omega (Y,X).$$ By Lemma \[identification1\], and it follows that $b\notin s^w_r(\omega _1,\check \omega _2)$. This shows that $p\le r$ when holds, and the result follows in the case $r\ge 1$.

Next assume that $r<1$. If holds for some $q>r$, then it follows by (real) interpolation between the cases and $${\operatorname{Op}}^2({\EuScript M}^{2,2}_{(\omega )}({\mathbf R^{2d}}))= {\mathscr I}_2(M^2_{(\omega _1)}({\mathbf R^{d}}),
M^2_{(\omega _2)}({\mathbf R^{d}}))$$ that holds for $r=1$ and some $q>1$. This contradicts the first part of the proof. If instead holds for some $p>r$, then it again follows by interpolation that holds for $r=1$ and some $p>1$, which again contradicts the first part of the proof. This shows that $p,q\le r$ if should hold. Furthermore, by Corollary it follows that $q\le 2$ when holds, and (1) follows.

It remains to prove (2). By [@GH1 Corollary 3.5] it follows that the result is true for trivial weights in the modulation space norms, and the result is carried over to the case with non-trivial weights by using lifting properties, established in [@GrochToft1]. The proof is complete.

Applications to the H[ö]{}rmander-Weyl calculus {#sec4}
===============================================

In this section we apply the results in the previous section to deduce Schatten-von Neumann properties in the H[ö]{}rmander-Weyl calculus. (See e.g. [@Le; @Toft4], Sections 18.4–18.6 in [@Ho1] and Subsection \[subsec1.8\] for approaches or notations.)

The following result is a consequence of Theorem 4.4 (1) in [@Toft4] in the case $p\ge 1$, while the latter result do not touch the case when $p<1$.

\[thm:WeylHorm1\] Let $p\in (0,2]$, $g$ be feasible on $W$, and let $m$ be a $g$-continuous weight on $W$ such that $m\in L^p(W)$. Then $S(m,g)\subseteq s_p^w(W)$.

We need some preparations for the proofs. First we recall that for any feasible metric $g$ and any $X\in W$, there are symplectic coordinates, and numbers $$0<\lambda _d(X)\le \dots \le \lambda _1(X)\le 1$$ such that $$g_X(Y) = \sum _{k=1}^d \lambda _k(X)(y_k^2+\eta _k^2),
\quad
g_X^\sigma (Y) = \sum _{k=1}^d \lambda _k(X)^{-1}(y_k^2+\eta _k^2),$$ where $Y=(y,\eta )\in W$ in these coordinates. The intermediate metric $$g^{{}_0}_X(Y) = \sum _{k=1}^d (y_k^2+\eta _k^2)$$ is symplectically invariant defined and is *symplectic*, i.e. $(g^{{}_0})^\sigma =g^{{}_0}$.

We have the following lemma.

\[prop:sumOpsSchatten\] Let $p\in (0,1]$, ${\mathscr H}_1$ and ${\mathscr H}_2$ be Hilbert spaces, and let $T_j\in {\mathcal B}({\mathscr H}_1,{\mathscr H}_2)$, $j\ge 1$. Then $${\Vert T\Vert _{{\mathscr I}_p({\mathscr H}_1,{\mathscr H}_2)}}
\le
\left (\sum _{k=1}^\infty {\Vert T_k\Vert _{{\mathscr I}_p({\mathscr H}_1,{\mathscr H}_2)}}^p \right )^{1/p},
\qquad
T=\sum _{k=1}^\infty T_k ,$$ provided the right-hand side makes sense as an element in ${\mathcal B}({\mathscr H}_1,{\mathscr H}_2)$.

We refer to [@Pe Appendix 1.1] for the proof of Lemma \[prop:sumOpsSchatten\].

\[prop:sumOpsSchatten2\] Let $p\in (0,1]$, $t\in \mathbf R$ and $a\in {\mathscr S}'({\mathbf R^{2d}})$. Also let $\{ÊÊ{\varphi}_j Ê\} _{j\in I}$ be a sequence in ${\mathscr S}({\mathbf R^{2d}})$ such that $0\le {\varphi}_j\le 1$ for every $j$ and $$\sum _{j\in I}{\varphi}_j =1.$$ Then $$\label{stpSumEst}
{\Vert a\Vert _{s_{t,p}({\mathbf R^{2d}})}}
\le
\left (  \sum _{j\in I}  {\Vert {\varphi}_ja\Vert _{s_{t,p}({\mathbf R^{2d}})}}^p \right )^{1/p}.$$

Let $a_j={\varphi}_ja$, $I_k\subseteq I$, $k\ge 1$ be a sequence of increasing and finite sets such that $\bigcup _{k=1}^\infty I_k=I$, and set $b_k=\sum _{j\in I_k}a_j$. We may assume that $$\label{Eq:SumajEst}
\sum _{j\in I}  {\Vert a_j\Vert _{s_{t,p}}}^p<\infty ,$$ since otherwise there is nothing to prove.

If $k_1\le k_2$, then Lemma \[prop:sumOpsSchatten\] gives $$\label{Eq:bkEsts}
{\Vert b_{k_2}-b_{k_1}\Vert _{s_{t,p}}}^p \le \sum _{j\in I_{k_2}\setminus I_{k_1}}
{\Vert a_j\Vert _{s_{t,p}}}^p
\quad\text{and}\quad
{\Vert b_k\Vert _{s_{t,p}}}^p \le \sum _{j\in I_k} {\Vert a_j\Vert _{s_{t,p}}}^p.$$ By we get ${\Vert b_{k_2}-b_{k_1}\Vert _{s_{t,p}}}\to 0$ as $k_1,k_2\to \infty$, and by completeness, there is a unique element $b\in {s_{t,p}}$ such that ${\Vert b_k-b\Vert _{s_{t,p}}}\to 0$ as $k\to \infty$.

Since $s_{t,p}\subseteq s_{t,2}=L^2$, we get $b\in L^2$ and $b_k\to b$ in $L^2$ as well when $k\to \infty$. Furthermore, since $|b_k|\le |a|$ and $b_k\to a$ pointwise, Beppo Levi’s theorem gives that $a\in L^2$ and $a=b$. The result now follows by letting $k$ tends to $\infty$ in .

Since $S(m,g)\subseteq
S(m,g^{{}_0})$ when $g^{{}_0}$ is the symplectic metric of $g$, and that $m$ is $g^{{}_0}$-continuous when $m$ is $g$-continuous, we may assume that $g$ is symplectic. (Cf. [@Toft4].)

Let $c>0$, $C>0$, $I$, $X_j$, $U_j$ and ${\varphi}_j$ be chosen as in Remarks 2.3 and 2.4 in [@Toft4], except that we may assume that $c>0$ and $C>0$ was chosen such that $$\begin{gathered}
C^{-1}m(X)\le m(Y)\le Cm(X)
\quad \text{and}\quad
C^{-1}g_X\le g_Y\le Cg_X
\\[1ex]
\text{when}\quad g_Y(X-Y)<2c\end{gathered}$$ Also set $a_j={\varphi}_ja$. Then ${\operatorname{supp}}a_j\subseteq U_j$, and the set of $a_j$ is bounded in $S(m,g)$. We first estimate ${\Vert a_j\Vert _{s_p^w}}$ for a fixed $j\in I$.

Let $B_r(X_0)$ denotes the open ball with center at $X_0$ and radius $r$, $\Phi \in C_0^\infty (B_1(0))\setminus 0$ be such that $0\le \Phi \le 1$, and choose symplectic coordinates $X=(x,\xi )\in {\mathbf R^{d}}\times {\mathbf R^{d}}\simeq
{\mathbf R^{2d}}$ such that $g_{X_j}$ takes the form $$g_{X_j}(X) = g_{X_j}(x,\xi ) = |X|^2=|x|^2 + |\xi |^2,$$ in these coordinates. Then for every multi-index $\alpha$, we have $\partial _g^\alpha = \partial _x^{\alpha _1}\partial _\xi ^{\alpha _2}$ for some $\alpha _1$ and $\alpha _2$ such that $\alpha _1+\alpha _2=\alpha$, and the support of $a_j$ is contained in $B_c(X_j)$. Let $N\ge 0$ be an integer such that $Np>d$. By Theorem 3.1 in [@GaSa] and Theorem \[thmOpSchatten\] we get $$\begin{gathered}
\label{ajSchattenEst1}
{\Vert a_j\Vert _{s_p^w}}^p \le C_1\iiiint | V_\Phi a_j(x,\xi ,\eta ,y)|^p\, dxd\xi d\eta dy
\\[1ex]
\le C_2\sum _{|\alpha +\beta|=2N} \iiiint | V_{\partial ^\beta \Phi} (\partial
^\alpha a_j)(x,\xi ,\eta ,y){\langle (y,\eta )\rangle}^{-2N}|^p\, dxd\xi d\eta dy
\\[1ex]
\le C_3\sum _{|\alpha \le 2N} \iiiint | V_\Phi (\partial
^\alpha a_j)(x,\xi ,\eta ,y)|^p{\langle (y,\eta )\rangle}^{-2Np}\, dxd\xi d\eta dy,\end{gathered}$$ for some constants $C_1,\dots ,C_3$ which are independent of $j\in I$.

We shall estimate the last integrand in . Let $\chi _{0,j}$ be the characteristic function of $U_j=B_c(X_j)$, and let $\chi _j$ be the characteristic function of $U_j=B_{1+c}(X_j)$. Then $$|(\partial ^\alpha a_j)(X)|\le C_{1,\alpha}m(X)\chi _{0,j}(X)
\le C_{2,\alpha}m(X_j)\chi _{0,j}(X),$$ for some constants $C_{1,\alpha}$ and $C_{2,\alpha}$, which only depend on $\alpha$. Here the last step follows from the fact that $m$ is $g$-continuous. This gives $$\begin{gathered}
| V_\Phi (\partial ^\alpha a_j)(x,\xi ,\eta ,y)|
\le
(2\pi )^{-d/2}\int _{{\mathbf R^{2d}}} |(\partial ^\alpha a_j)(Z-X)|\Phi (Z)\, dZ
\\[1ex]
\le
C_{3,\alpha}m(X_j)\int _{{\mathbf R^{2d}}} |\chi _{0,j}(Z-X)|\Phi (Z)\, dZ
\le
C_{4,\alpha}m(X_j)\chi _j(X)
\\[1ex]
=
C_{5,\alpha}\left (m(X_j)^p\int {\varphi}_j(Z)\, dZ \right )^{1/p} \chi _j(X)
\\[1ex]
\le
C_{6,\alpha}\left ( \int m(Z)^p{\varphi}_j(Z)\, dZ \right )^{1/p} \chi _j(X),\end{gathered}$$ for some constants $C_{3,\alpha},\dots ,C_{6,\alpha}$, which only depend on $\alpha$.

By combining the last estimate with , we get $$\begin{gathered}
{\Vert a_j\Vert _{s_p^w}}^p\le C_1 \left (  \iint \chi _j(X)^p{\langle Y\rangle}^{-2Np}\, dXdY  \right )
\cdot  \left ( \int m(Z)^p{\varphi}_j(Z)\, dZ  \right )
\\[1ex]
\le
C_2\int m(Z)^p{\varphi}_j(Z)\, dZ,\end{gathered}$$ for some constants $C_1$ and $C_2$, which are independent of $j$.

A combination of the last estimate and Corollary \[prop:sumOpsSchatten2\] now gives $${\Vert a\Vert _{s_p^w}}^p \le \sum _j {\Vert a_j\Vert _{s_p^w}}^p
\lesssim \sum _j \int m(Z)^p{\varphi}_j(Z)\, dZ = {\Vert m\Vert _{L^p}}^p,$$ and the result follows.

By similar arguments as in the proof of [@Toft4 Theorem 2.11], Theorem \[thm:WeylHorm1\] gives the following results involving suitable Sobolev type spaces, introduced in by Bony and Chemin in [@BoCh] (see also [@Le]). The details are left for the reader. Here recall that the Riemannian metric $g$ on $W$ is called *split* if there are symplectic coordinates such that $$g_X(y,\eta ) = g_X(y,-\eta ),\qquad X\in W,\ Y=(y,\eta )\in W,$$ in these coordinates.

\[thm:WeylHorm3\] Let $t=1/2$, $a\in {\mathscr S}'(W)$, $p\in (0,\infty ]$, $g$ be strongly feasible on $W$, and let $m,m_1,m_2$ be $g$-continuous and $(\sigma ,g)$-temperate weights on $W$ such that $m_2m/m_1\in L^p(W)$. Then $S(m,g)\subseteq s_{t,p}(m_1,m_2,g)$.

Furthermore, if in addition $g$ is split, then $S(m,g)\subseteq s_{t,p}(m_1,m_2,g)$ holds for general $t\in \mathbf R$.

We note that [@Toft4 Theorem 2.11] covers Theorem \[thm:WeylHorm3\] when $p\ge 1$. We also note that the assumption $a\in S(m,g)$ is missing, and that $s_{t,p}^w$ and $s_{t,\sharp}^w$ should be $s_{t,p}$ and $s_{t,\sharp}$, respectively, in (3) and (4) in [@Toft4 Theorem 2.11].

Applications to compactly supported Schatten-von Neumann symbols {#sec5}
================================================================

In this section we introduce a subset $s_{t,p}^q({\mathbf R^{2d}})$ of $s_{t,p}({\mathbf R^{2d}})$ when $p,q\in (0,\infty ]$. We show that the set of compactly supported elements in $s_{t,p}^q({\mathbf R^{2d}})$ with $q=\min (p,1)$ and $q=1$ is a subspace and superspace, respectively, of compactly supported elements in $\mathscr F L^p$. This result goes back to [@Toft0; @Toft1] in the case $p\ge 1$ and $t=1/2$. The proof is based on Theorem \[thmOpSchatten\] and certain characterizations given here, which might be of independent interests.

First we make the following definition. Here ${\operatorname{ON}}_d$ is the set of all orthonormal sequences in $L^2({\mathbf R^{d}})$.

Let $t\in \mathbf R$ and $p,q\in (0,\infty ]$. Then $s_{t,p}^q({\mathbf R^{2d}})$ consists of all $a\in {\mathscr S}'({\mathbf R^{2d}})$ of the form $$a=\sum _{j=0}^\infty \lambda _j W^t_{f_j,g_j},$$ for some non-negative and non-increasing sequence $\{\lambda _j \} _{j=0}^\infty \in \ell ^p(\mathbf N)$, and some $\{ f_j \} _{j=0}^\infty \in {\operatorname{ON}}_d$ and $\{ g_j \}
_{j=0}^\infty \in {\operatorname{ON}}_d$ which at the same time are bounded in $M^{2q}({\mathbf R^{d}})$. For any $a\in s_{t,p}^q({\mathbf R^{2d}})$ we set $${\Vert a\Vert _{s_{t,p}^q}} \equiv {\Vert \{ \lambda _j {\Vert f_j\Vert _{M^{2q_0}}}
{\Vert g_j\Vert _{M^{2q_0}}} \} _{j=0}^\infty \Vert _{\ell ^p}},\qquad q_0=\min (1,q).$$

In the following lemma we present some basic facts for $s_{t,p}^q({\mathbf R^{2d}})$.

\[basicLemmaspp\] Let $p,q,p_j,q_j\in (0,\infty ]$, $j=0,1,2$ be such that $p_1\le p_2$, $q_1\le q_2$ and $q_0\ge 1$. Then the following is true:

1.  $s_{t,p}^{q_0}({\mathbf R^{2d}})=s_{t,p}({\mathbf R^{2d}})$ and ${\Vert a\Vert _{s_{t,p}^{q_0}}}
    ={\Vert a\Vert _{s_{t,p}}}$ when $a \in s_{t,p}({\mathbf R^{2d}})$;

2.  ${\mathscr S}({\mathbf R^{2d}})\subseteq s_{t,p}^q({\mathbf R^{2d}})$;

3.  $s_{t,p_1}^{q_1}({\mathbf R^{2d}})\subseteq s_{t,p_2}^{q_2}({\mathbf R^{2d}})$;

4.  if in addition $t=1/2$, then $a\in s_{t,p}^{q}({\mathbf R^{2d}})$, if and only if ${\mathscr F}_\sigma a\in s_{t,p}^{q}({\mathbf R^{2d}})$, and $${\Vert {\mathscr F}_\sigma a\Vert _{s_{t,p}^q}} = {\Vert a\Vert _{s_{t,p}^q}} .$$

The assertion (1) follows from the fact that $L^2({\mathbf R^{d}})$ is continuously embedded in $M^{2q_0}({\mathbf R^{d}})$, since $2q_0\ge 2$, (2) follows from Proposition \[SchattenExp\] below, and (3) is a straight-forward consequence of the definitions.

Finally, (4) follows from the facts that ${\mathscr F}_\sigma W_{f,g}=W_{\check f,g}$ and that ${\Vert \check f\Vert _{M^{2q}}}\asymp {\Vert f\Vert _{M^{2q}}}$. This gives the result.

The following result characterizes the set of compactly supported elements in $s_{t,p}({\mathbf R^{2d}})$.

\[CompSuppSchatten\] Let $t\in \mathbf R$ and $p,q\in (0,\infty ]$ be such that $r\le q$. Then $$\begin{gathered}
\label{CompSuppSchattenEq}
s^q_{t,q}({\mathbf R^{2d}})\bigcap \mathscr E'({\mathbf R^{2d}})
\subseteq \mathscr FL^q({\mathbf R^{2d}}) \bigcap \mathscr E'({\mathbf R^{2d}})
\\[1ex]
= M^{p,q}({\mathbf R^{2d}})\bigcap \mathscr E'({\mathbf R^{2d}})
\subseteq s_{t,q}({\mathbf R^{2d}})\bigcap \mathscr E'({\mathbf R^{2d}}).\end{gathered}$$

We need some preparations for the proof. First we recall the following facts for the harmonic oscillator $H=H_d=|x|^2-\Delta$ on ${\mathscr S}'({\mathbf R^{d}})$. We omit the proof since the result is a special case of [@BoTo Theorem 3.5].

\[LemmaHarmOscBij\] Let $p,q\in [1,\infty ]$ and let $\omega ,\vartheta _N\in {\mathscr P}({\mathbf R^{2d}})$ be such that $\vartheta _N(x,\xi )= {\langle (x,\xi )\rangle}^N$, where $N$ is an integer. Then $H_d^N$ is a homeomorphism on ${\mathscr S}({\mathbf R^{d}})$, on ${\mathscr S}'({\mathbf R^{d}})$, and from $M^{p,q}_{(\vartheta _{2N} \omega )}({\mathbf R^{d}})$ to $M^{p,q}_{(\omega )}({\mathbf R^{d}})$.

The next result concerns Schwartz kernels of linear operators.

\[SchwartzKernels\] Let $T$ be a linear and continuous operator from $L^2({\mathbf R^{d_1}})$ to $L^2({\mathbf R^{d_2}})$ and such that the kernels of $T^*\circ T$ and $T\circ T^*$ belong to ${\mathscr S}({\mathbf R^{2d_1}})$ and ${\mathscr S}({\mathbf R^{2d_2}})$, respectively. Then the kernel of $T$ belongs to ${\mathscr S}({\mathbf R^{d_2+d_1}})$.

The result should be available in the literature. In order to be self-contained we here present a proof, obtained in collaboration with A. Holst at Lund University, Sweden.

Let $N\in \mathbf N$, $\omega _{j,r}(x,\xi )= {\langle (x,\xi )\rangle}^{r}$ when $x,\xi \in
{\mathbf R^{d_j}}$ and $r\in \mathbf R$, and let $f_j\in {\mathscr S}({\mathbf R^{d_j}})$, $j=1,2$. Then $H^N\circ S\circ H^N$ is an operator with kernel in ${\mathscr S}$, if and only if $S$ is an operator with kernel in ${\mathscr S}$.

By the assumptions and Lemma \[LemmaHarmOscBij\] we obtain $$\begin{aligned}
{\Vert Tf_1\Vert _{L^2({\mathbf R^{d_2}})}}^2 &= ((T^*\circ T)f_1,f_1)_{L^2({\mathbf R^{d_1}})}
\lesssim {\Vert f_1\Vert _{M^2_{(\omega _{1,-2N})}}}^2
\intertext{and}
{\Vert T^*f_2\Vert _{L^2({\mathbf R^{d_1}})}}^2 &= ((T\circ T^*)f_2,f_2)_{L^2({\mathbf R^{d_2}})}
\lesssim {\Vert f_2\Vert _{M^2_{(\omega _{2,-2N})}}}^2.\end{aligned}$$ Hence $$T\in {\mathcal B}(M^2_{(\omega _{1,-2N})}({\mathbf R^{d_1}}),
L^2({\mathbf R^{d_2}}))
\quad \text{and} \quad T^* \in {\mathcal B}(M^2_{(\omega _{2,-2N})}
({\mathbf R^{d_2}}),L^2({\mathbf R^{d_1}})).$$

By duality it also follows that $T\in {\mathcal B}(L^2({\mathbf R^{d_1}}),
M^2_{(\omega _{2,2r})}({\mathbf R^{d_2}}))$, since the dual of $M^2_{(\omega _{j,-N})}({\mathbf R^{d_j}})$ equals $M^2_{(\omega _{j,N})}({\mathbf R^{d_j}})$ when the $L^2$ form is used (cf. e.g. [@Gc2 Theorem 11.3.6]).

By interpolation between these results we get $$T\in {\mathcal B}(M^2_{(\omega _{1,-N})}({\mathbf R^{d_1}}),
M^2_{(\omega _{2,N})}({\mathbf R^{d_2}})).$$ and since $$\bigcap _{r\in \mathbf R} M^2_{(\omega _{j,N})}({\mathbf R^{d_j}})
={\mathscr S}({\mathbf R^{d_j}})
\quad \text{and}\quad
\bigcup _{r\in \mathbf R} M^2_{(\omega _{j,r})}({\mathbf R^{d_j}})
={\mathscr S}'({\mathbf R^{d_j}})$$ when $j=1,2$ also in topological sense in view of e.g. [@Toft3], we obtain $T\in {\mathcal B}({\mathscr S}'({\mathbf R^{d_1}}) , {\mathscr S}({\mathbf R^{d_2}}))$. This implies that the kernel of $T$ belongs to ${\mathscr S}({\mathbf R^{d_2+d_1}})$.

The next result extends in several ways Lemma 4.1.2 in [@Toft0] and concerns suitable Wigner distribution expansions (see ).

\[SchattenExp\] Let $a\in {\mathscr S}({\mathbf R^{2d}})$, $t\in \mathbf R$, $p,q\in (0,\infty ]$ and let $H=|x|^2-\Delta$ be the harmonic oscillator on ${\mathbf R^{d}}$. Then $$\label{SymbolSpectralExpansion}
a=\sum _{j=0}^\infty \lambda _j W^t_{f_j,g_j},$$ for some non-negative and non-increasing $\{\lambda _j\} _{j=0}^\infty \subseteq
\mathbf R$, $\{ f_j \} _{j=0}^\infty \in {\operatorname{ON}}(L^2({\mathbf R^{d}}))$ and $\{ g_j \} _{j=0}^\infty \in {\operatorname{ON}}(L^2({\mathbf R^{d}}))$ such that $$\begin{gathered}
\lambda _j \ge 0,\quad f_j,g_j\in {\mathscr S}({\mathbf R^{d}}),\quad
j\ge 0,\notag
\intertext{and}
\sum _{j=0}^\infty \lambda _j^p {\Vert H^Nf_j\Vert _{M^q}}
{\Vert H^Ng_j\Vert _{M^q}}<\infty ,
\quad \text{when}\quad
N\ge 0.\label{SymbSpecExpEst}\end{gathered}$$

By the spectral theorem of compact operators, holds true for some non-negative and non-increasing sequence $\{\lambda _j\} _{j=0}^\infty$, and some $\{ f_j \} _{j=0}^\infty$ and $\{ g_j \} _{j=0}^\infty$ in ${\operatorname{ON}}_d$. Since $$e^{i(t-1/2){\langle D_\xi,D_x\rangle}}W^t_{f,g}=W_{f,g}$$ and $e^{i(t-1/2){\langle D_\xi,D_x\rangle}}$ is continuous on ${\mathscr S}({\mathbf R^{2d}})$, it suffices to consider the Weyl case, $t=1/2$.

First we assume that $T\equiv {\operatorname{Op}}^w(a)\ge 0$, giving that holds with $g_j=f_j$. The result is true for $p=1$ and $q=2$ in view of [@Toft0 Lemma 4.1.2]. Since the kernel of $T$ belongs to ${\mathscr S}$, Proposition \[SchwartzKernels\] shows that the kernel of $T_N = T^{1/2N}$ belongs to ${\mathscr S}$ for every $N\ge 1$. Furthermore, if $a_N$ is the Weyl symbol of $T_N$, then $a_N\in {\mathscr S}({\mathbf R^{2d}})$, and by straight-forward computations we get $$a_N = \sum _j \lambda _j^{1/2N}W_{f_j,f_j}.$$ By [@Toft0 Lemma 4.1.2] we get $$\sum _j \lambda _j^{1/2N}{\Vert H^Nf_j\Vert _{L^2}}^2<\infty$$ for every $N\ge 0$, and the result follows in the case $p>0$ and $q=2$.

Next assume that $q\in (0,2)$, and let $\omega _r ={\langle {\, \cdot \, }\rangle} ^r
\in {\mathscr P}({\mathbf R^{2d}})$, $q_0=(2q)/(2-q)$ and $N_0>dq_0$ be an integer. Then $\omega _{-2N_0}\in L^{q_0}({\mathbf R^{2d}})$, and H[ö]{}lder’s inequality gives $${\Vert H^Nf\Vert _{M^q}}\lesssim {\Vert \omega _{-2N_0}\Vert _{L^{q_0}}}
{\Vert H^Nf\Vert _{M^2_{(\omega _{2N_0})}}} \asymp {\Vert H^{N+N_0}f\Vert _{L^2}},$$ when $f\in {\mathscr S}({\mathbf R^{d}})$, and the result follows in this case from the case $q=2$.

If instead $q\ge2$, then ${\Vert H^Nf\Vert _{M^q}}\lesssim {\Vert H^Nf\Vert _{L^2}}$ for every admissible $f$, and the result again follows from the case when $q=2$.

The assertion therefore follows if in the case ${\operatorname{Op}}_t(a)\ge 0$.

Finally assume that $a\in {\mathscr S}({\mathbf R^{2d}})$ is general. The Weyl symbol of the operators $T^*\circ T$ and $T\circ T^*$ are given by $$b=\sum \lambda _j^2W_{f_j,f_j}\quad \text{and}\quad
c=\sum \lambda _j^2W_{g_j,g_j},$$ respectively, and belong to ${\mathscr S}({\mathbf R^{2d}})$, in view of Proposition \[SchwartzKernels\]. Since $T^*\circ T$ and $T\circ T^*$ are positive semi-definite, it follows from the first part of the proof that $$\sum _j \lambda _j^p{\Vert H^Nf_j\Vert _{M^q}}^2<\infty 
\quad \text{and}\quad
\sum _j \lambda _j^p{\Vert H^Ng_j\Vert _{M^q}}^2<\infty$$ hold for every $p,q\in (0,\infty ]$ and $N\ge 0$. The estimate now follows from these estimates and Cauchy-Schwartz inequality.

We also need the following result related to Theorem 3.1 in [@GaSa].

\[STFTWindows\] Let $p\in (0,2]$ and let $\omega ,\omega _1,\omega _2\in {\mathscr P}_E({\mathbf R^{2d}})$ be such that $$\omega (X_1-X_2)\lesssim \omega _1(X_1)\omega _2(X_2),\qquad X_1,X_2\in {\mathbf R^{2d}}.$$ Then the map $(f,\phi )\mapsto V_\phi f$ is continuous from $M^p_{(\omega _1)}({\mathbf R^{d}})
\times M^p_{(\omega _2)}({\mathbf R^{d}})$ to $L^p_{(\omega )}({\mathbf R^{2d}})$, and $$\label{NormsEsts}
{\Vert V_\phi f\Vert _{L^p_{(\omega )}}} \le C {\Vert f\Vert _{M^p_{(\omega _1)}}}
{\Vert \phi\Vert _{M^p_{(\omega _2)}}},$$ where the constant $C$ is independent of $f\in M^p_{(\omega _1)}({\mathbf R^{d}})$ and $\phi \in M^p_{(\omega _2)}({\mathbf R^{d}})$.

First assume that $p\le 1$. Let $\psi \in \Sigma _1({\mathbf R^{d}})$, ${\varepsilon}>0$, $\Lambda ={\varepsilon}{\mathbf Z^{2d}}$, $\{ c({\boldsymbol j})\} _{{\boldsymbol j}\in \Lambda}\in \ell
^p_{(\omega _1)}(\Lambda )$ and $\{ d({\boldsymbol k})\} _{{\boldsymbol k}\in \Lambda}
\in \ell ^p_{(\omega _2)}(\Lambda )$ be chosen such that $$f(x) = \sum _{j,\iota \in {\varepsilon}{\mathbf Z^{d}}}c(j,\iota )\psi (x-j)e^{i{\langle x,\iota\rangle}}
\quad \text{and}\quad
\phi (x) = \sum _{k,\kappa \in {\varepsilon}{\mathbf Z^{d}}}d(k,\kappa )\psi (x-k)e^{i{\langle x,\kappa\rangle}}$$ (cf. Proposition \[ConseqThmS\]).

By straight-forward computations it follows that $$V_\phi f(X) = \sum _{{\boldsymbol j},{\boldsymbol k}\in \lambda} c({\boldsymbol j})\overline{d({\boldsymbol k})}
\Psi (X+{\boldsymbol k}-{\boldsymbol j}) R_{{\boldsymbol j},{\boldsymbol k}}(X),$$ where $\Psi =V_\psi \psi \in \Sigma _1({\mathbf R^{2d}})$ and $R_{{\boldsymbol j},{\boldsymbol k}}$ is a function of exponential type such that $|R_{{\boldsymbol j},{\boldsymbol k}}| =1$, for every ${\boldsymbol j}$ and ${\boldsymbol k}$. This gives $$\begin{gathered}
{\Vert V_\phi f\Vert _{L^p_{(\omega )}}} ^p
\le
\sum _{{\boldsymbol j},{\boldsymbol k}\in \Lambda} |c({\boldsymbol j})|^p |d({\boldsymbol k})|^p
{\Vert \Psi ({\, \cdot \, }+{\boldsymbol k}-{\boldsymbol j})\omega \Vert _{L^p}}^p
\\[1ex]
\le
\sum _{{\boldsymbol j},{\boldsymbol k}\in \Lambda} |c({\boldsymbol j})|^p |d({\boldsymbol k})|^p
{\Vert \Psi v \Vert _{L^p}}^p\omega ({\boldsymbol j}-{\boldsymbol k})^p
\\[1ex]
\lesssim
\sum _{{\boldsymbol j},{\boldsymbol k}\in \Lambda} |c({\boldsymbol j})\omega _1({\boldsymbol j})|^p
|d({\boldsymbol k})\omega _2({\boldsymbol j})|^p
\asymp {\Vert f\Vert _{M^p_{(\omega _1)}}}{\Vert \phi\Vert _{M^p_{(\omega _2)}}},\end{gathered}$$ when $v\in {\mathscr P}_E({\mathbf R^{2d}})$ is chosen such that $\omega$ is $v$-moderate. Here the first inequality follows from the fact that $p\le 1$ and the last inequality follows from the assumptions. This gives the result in the case $p\le 1$.

A slight modification of the proof of (2.8) in [@Toft11] gives the result in the remaining case where $p\in [1,2]$. The details are left for the reader.

The next result concerns extensions of certain convolution relations in [@Toft0; @Toft1] between Schatten-von Neumann symbols and Lebesgue spaces to the case when the Lebesgue parameters are allowed to be smaller than $1$.

\[SchattenConv\] Let $p\in (0,1]$ and $t=1/2$. Then the map $(a,b)\mapsto a*b$ is continuous from $s^p_{t,p}({\mathbf R^{2d}})\times s^p_{t,p}({\mathbf R^{2d}})$ to $L^p({\mathbf R^{2d}})$.

Let $a,b\in s^p_{t,p}$. Since $s^p_{t,p}\subseteq s_{t,1}$, it follows that $a*b$ is well-defined and belongs to $L^1$, in view of Theorem 2.1 in [@Toft1].

Now let $$\begin{gathered}
\{ \lambda _j \} _{j=0}^\infty \in \ell ^p (\mathbf N),
\quad
\{ \mu _j \} _{j=0}^\infty \in\ell ^p (\mathbf N),\quad \{ f_{l,j} \} _{j=0}^\infty \in {\operatorname{ON}}(L^2({\mathbf R^{d}}))
\intertext{and}
\{ g_{l,j} \} _{j=0}^\infty \in {\operatorname{ON}}(L^2({\mathbf R^{d}})),\quad l=1,2,\end{gathered}$$ be such that $\lambda _j\ge 0$ and $\mu _j\ge 0$ for every $j\ge 0$, $$\sup _{j,l}{\Vert f_{l,j}\Vert _{M^{2p}}}<\infty
\quad \text{and}\quad
\sup _{j,l}{\Vert g_{l,j}\Vert _{M^{2p}}}<\infty ,$$ and $$a=\sum _{j=0}^\infty \lambda _j W_{f_{1,j},f_{2,j}}
\quad \text{and}\quad
b=\sum _{j=0}^\infty \mu _j W_{g_{1,j},g_{2,j}} .$$ Then $$\begin{gathered}
\label{convLebSchattEst}
{\Vert a*b\Vert _{L^p}}^p = \int \left | \sum _{j,k} \lambda _j\mu _k
(W_{f_{1,j},f_{2,j}}*W_{g_{1,k},g_{2,k}} )(X)\right |^p \, dX
\\[1ex]
\le
\sum _{j,k} \lambda _j^p\mu _k^p
{\Vert W_{f_{1,j},f_{2,j}}*W_{g_{1,k},g_{2,k}}\Vert _{L^p}}^p.\end{gathered}$$

By straight-forward computations we get $$|(W_{f_{1,j},f_{2,j}}*W_{g_{1,k},g_{2,k}} ) (x,\xi )|
=
C|(V_{\check f_{2,j}}g_{1,k})(x,\xi )(V_{\check f_{1,j}}g_{2,k})(x,\xi )|,$$ for some constant $C$. Hence Cauchy-Schwartz inequality and Lemma \[STFTWindows\] give $$\begin{gathered}
{\Vert W_{f_{1,j},f_{2,j}}*W_{g_{1,k},g_{2,k}}\Vert _{L^p}} =
{\Vert V_{\check f_{2,j}}g_{1,k}\cdot V_{\check f_{1,j}}g_{2,k}\Vert _{L^p}}
\\[1ex]
\le
{\Vert V_{\check f_{2,j}}g_{1,k}\Vert _{L^{2p}}}{\Vert V_{\check f_{1,j}}g_{2,k}\Vert _{L^{2p}}}
\lesssim {\Vert f_{1,j}\Vert _{M^{2p}}}{\Vert f_{2,j}\Vert _{M^{2p}}}{\Vert g_{1,k}\Vert _{M^{2p}}}{\Vert g_{2,k}\Vert _{M^{2p}}}.\end{gathered}$$

By inserting this into we get $$\begin{gathered}
{\Vert a*b\Vert _{L^p}} \lesssim \left ( \sum _{j,k} \lambda _j^p\mu _k^p
{\Vert f_{1,j}\Vert _{M^{2p}}}^p {\Vert f_{2,j}\Vert _{M^{2p}}}^p
{\Vert g_{1,k}\Vert _{M^{2p}}}^p {\Vert g_{2,k}\Vert _{M^{2p}}}^p \right )^{1/p}
\\[1ex]
= {\Vert a\Vert _{s_p^p}}{\Vert b\Vert _{s_p^p}},\end{gathered}$$ and the result follows.

The equality and the last embedding in follow from [@Toft12 Proposition 4.3] and Theorem \[thmOpSchatten\]. The first embedding in follows from Corollary 2.12 in [@Toft1] in the case $p\ge 1$. It remains to prove the first embedding in in the case $p<1$.

Therefore, assume that $p<1$, let $a\in {\mathscr E}'({\mathbf R^{2d}})$ and choose ${\varphi}\in C_0^\infty ({\mathbf R^{2d}})$ such that ${\varphi}=1$ on ${\operatorname{supp}}a$. Then ${\mathscr F}_\sigma
{\varphi}\in s_{t,p}^p({\mathbf R^{2d}})$ by Proposition \[SchattenExp\]. Hence Lemma \[basicLemmaspp\] and Proposition \[SchattenConv\] give $$\begin{gathered}
{\Vert a\Vert _{{\mathscr F}L^p}} = {\Vert {\varphi}a\Vert _{{\mathscr F}L^p}} \asymp {\Vert ({\mathscr F}_\sigma {\varphi})
* ({\mathscr F}_\sigma a)\Vert _{L^p}}
\\[1ex]
\lesssim {\Vert {\mathscr F}_\sigma {\varphi}\Vert _{s_{t,p}^p}} {\Vert {\mathscr F}_\sigma a\Vert _{s_{t,p}^p}}
\lesssim {\Vert a\Vert _{s_{t,p}^p}},\end{gathered}$$ which gives the result.

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