# Factorizations for quasi-Banach time-frequency spaces and Schatten classes

Divyang G. Bhimani  Department of Mathematics, Indian Institute of Science
Education and Research, Pune, India<EMAIL_ADDRESS>and
Joachim Toft  Department of Mathematics, Linnæus University, Växjö, Sweden
<EMAIL_ADDRESS>
###### Abstract.

We deduce factorization properties for Wiener amalgam spaces $W\\!\\!L^{p,q}$,
an extended family of modulation spaces $M(\omega,\mathscr{B})$, and for
Schatten symbols $s_{p}^{w}$ in pseudo-differential calculus under e. g.
convolutions, twisted convolutions and symbolic products. Here
$M(\omega,\mathscr{B})$ can be any quasi-Banach Orlicz modulation space. For
example we show that $W\\!\\!L^{1,r}*W\\!\\!L^{p,q}=W\\!\\!L^{p,q}$ and
$W\\!\\!L^{1,r}{\text{\footnotesize$\\#$}}s_{p}^{w}=s_{p}^{w}$ when
$r\in(0,1]$, $r\leq p,q<\infty$. In particular we improve Rudin’s identity
$L^{1}*L^{1}=L^{1}$.

###### Key words and phrases:

quasi-Banach spaces, approximate identity, algebras, modules, Wiener amalgam
spaces, modulation spaces, Schatten-von Neumann

###### 2010 Mathematics Subject Classification:

42A85, 42B35 (primary), 42B37 (secondary).

###### 2010 Mathematics Subject Classification:

46A16, 46Hxx, 42B35, 35S05, 47B10

##  0\. Introduction

Several issues in science and technology concern topological algebras and
modules (see e. g. [4, 6, 25] and the references therein). In this context a
relevant and fascinating question with an algebraic flavor arise quite
naturally, wether the algebra or module under consideration possess the so-
called factorization property. That is whether they are factorization algebras
and factorization modules, or not. More specifically, consider the pair
$(\mathcal{B},\mathcal{M},\,\cdot\,)$ of algebras and modules, where
$\mathcal{B}$ is a topological algebra, $\mathcal{M}$ is a topological (right)
module over $\mathcal{B}$ with multiplication $\cdot$ between elements in
$\mathcal{B}$ and $\mathcal{M}$. (See Sections 1, 4 or [32] for notations.) Is
it then true that for arbitrary elements $\phi\in\mathcal{B}$ and
$f\in\mathcal{M}$, there are elements $\phi_{j}\in\mathcal{B}$ and
$f_{0}\in\mathcal{M}$ such that

$\phi=\phi_{1}\phi_{2}\quad\text{and}\quad f=\phi_{0}\cdot f_{0}?$

Evidently, if $\mathcal{B}$ is unital, i. e. it contains the unitary element
$\varrho$, then

$\phi=\varrho\phi\quad\text{and}\quad f=\varrho\cdot f,$

giving the factorization property. On the other hand, if such unitary element
is missing, the question on factorization property becomes more subtle and
challenging, and less trivial.

As examples, let $p\in[1,\infty)$, and consider the pairs of Banach algebras
and modules

$(L^{\infty}(\mathbf{R}^{d}),L^{p}(\mathbf{R}^{d}),\,\cdot\,),\quad(L^{1}(\mathbf{R}^{d}),L^{p}(\mathbf{R}^{d}),*),\quad\text{and}\quad(\mathscr{I}_{2},\mathscr{I}_{\infty},\circ).$
(0.1)

Here $\mathscr{I}_{2}$ and $\mathscr{I}_{\infty}$ are the sets of Hilbert-
Schmidt and continuous operators on $L^{2}(\mathbf{R}^{d})$, respectively, and
$\circ$ denotes the operator composition. We also note that Hölder’s and
Young’s inequalities guarantee the Banach algebra and module properties for
the first two pairs in (0.1).

Then the first pair in (0.1) possess the factorization property because the
multiplicative identity $1$ is an element in the Banach algebra
$L^{\infty}(\mathbf{R}^{d})$.

For the second pair in (0.1), there is no identity element in the algebra
$(L^{1}(\mathbf{R}^{d}),*)$. However, despite this fact we indeed have

$L^{1}(\mathbf{R}^{d})*L^{p}(\mathbf{R}^{d})=L^{p}(\mathbf{R}^{d})$ (0.2)

when $1\leq p<\infty$. Thereby it follows that this pair possess the
factorization property.

Filter$\phi$$f_{\operatorname{in}}$
$f_{\operatorname{out}}=f_{\operatorname{in}}*\phi$

Picture 0.1 What kind of outgoing signals $f_{\operatorname{out}}$ can be
constructed from insignals $f_{\operatorname{out}}$ and filters with functions
$\phi$?

In signal analysis, a related question concerns what kind of outgoing signals
$f_{\operatorname{out}}$ belonging to e. g. $L^{1}$, can be obtained from
stationary filters. (See Picture 0.1.) Suppose for example that we can choose
filters with filter functions $\phi$ and ingoing signal
$f_{\operatorname{in}}$ being any $L^{1}$ functions. Then the answer on this,
in a pure theoretical framework, is that any $L^{1}$ function can be obtained
as outgoing signal, in view of (0.2). Evidently, in reality we may not choose
filter functions and ingoing signals as any $L^{1}$ functions.

For the third pair in (0.1) we have

$\mathscr{I}_{2}\circ\mathscr{I}_{\infty}=\mathscr{I}_{2}\subsetneq\mathscr{I}_{\infty},$

which shows that it cannot possess the factorization property. On the other
hand, by the swap of the rules of $\mathscr{I}_{2}$ and $\mathscr{I}_{\infty}$
in (0.1) into the pair $(\mathscr{I}_{\infty},\mathscr{I}_{2},\circ)$, i. e.
letting $\mathscr{I}_{2}$ be a module over the ring $\mathscr{I}_{\infty}$, we
have

$\displaystyle\mathscr{I}_{\infty}\circ\mathscr{I}_{2}$
$\displaystyle=\mathscr{I}_{2}.$ This is trivially true because because
$\mathscr{I}_{\infty}$ is unital. Less trivial is that indeed
$\displaystyle\mathscr{I}_{\sharp}\circ\mathscr{I}_{2}$
$\displaystyle=\mathscr{I}_{2},$ or more generally,
$\displaystyle\mathscr{I}_{\sharp}\circ\mathscr{I}_{p}$
$\displaystyle=\mathscr{I}_{p},\quad p\in[0,\infty),$ (0.3)

since $\mathscr{I}_{\sharp}$ is not unital. That is,
$(\mathscr{I}_{\sharp},\mathscr{I}_{p},\circ)$ possess factorization property.
Here $\mathscr{I}_{p}$ is the space of Schatten-von Neumann operators and
$\mathscr{I}_{\sharp}$ is the space of compact operators on
$L^{2}(\mathbf{R}^{d})$.

In the paper we deduce factorization properties for certain quasi-Banach
spaces under various kinds of multiplications.

In Section 2 we consider such factorizations for weighted versions of Wiener-
amalgam spaces $W\\!\\!L^{p,q}(\mathbf{R}^{d})$, $p,q\in(0,\infty]$, often
denoted by $W(L^{p}(\mathbf{R}^{d}),L^{q}(\mathbf{R}^{d}))$ in the literature,
with $L^{p}(\mathbf{R}^{d})$ as local component and $L^{q}(\mathbf{R}^{d})$ as
global component. That is, $W\\!\\!L^{p,q}(\mathbf{R}^{d})$ consists of all
Lebesgue measurable functions $f$ with finite quasi-norm

$\displaystyle\|f\|_{W\\!\\!L^{p,q}}$
$\displaystyle\equiv\|a_{f}\|_{\ell^{q}(\mathbf{Z}^{d})},$ (0.4)
$\displaystyle a_{f}(k)$ $\displaystyle=\|f\|_{L^{p}(k+Q)},\
k\in\mathbf{Z}^{d},\ Q=[0,1]^{d}$

(see e. g. [22, 24, 28]).

For $W\\!\\!L^{1,r}(\mathbf{R}^{d})$, the usual mollifiers become bounded
approximate identities under convolutions (see Example 1.7). This leads to the
following two theorems on factorizations of Wiener-amalgam and classical
modulation spaces. The first result is essentially Theorems 2.2 in Section 2
after restricting ourselves to allow only trivial weights.

######  Theorem 0.1.

Let $r\in(0,1]$, $p\in[1,\infty)$ and $q\in[r,\infty)$. Then

$W\\!\\!L^{1,r}*W\\!\\!L^{p,q}=W\\!\\!L^{p,q}.$ (0.5)

The next (second) result is essentially Theorem 3.3 in Section 3, after
restricting ourselves to allow only trivial weights and to classical
modulation spaces. Here we remark that Theorem 3.3 deals with factorizations
for a large family of quasi-Banach modulation spaces. (See Definition 1.15.)

######  Theorem 0.2.

Suppose $r\in(0,1]$ and $p,q\in[r,\infty)$. Then

$W\\!\\!L^{1,r}*M^{p,q}=M^{p,q}.$ (0.6)

Theorem 0.1 remains the same after the convolution $*$ is replaced by the
twisted convolution $*_{\sigma}$. (See Theorem 2.5.)

If $r=1$ and $q=p$ in Theorems 0.1 and 0.2, we recover Rudn’s identity

$\displaystyle L^{1}*L^{p}$ $\displaystyle=L^{p},$ (0.7) as well as reaching
$\displaystyle L^{1}*M^{p,q}$ $\displaystyle=M^{p,q},$ (0.8) which can be
obtained by straight-forward applications of Hewitt’s factorization result
[30, (2.5) Theorem].
If instead $r\leq 1$ but still supposing that $q=p$ in Theorem 0.1, then we
obtain $\displaystyle W\\!\\!L^{1,r}*L^{p}$ $\displaystyle=L^{p},\qquad r\leq
1.$ (0.9)

which is a strict improvement of (0.7), because $W\\!\\!L^{1,r}\subsetneq
L^{1}$ when $r<1$ (with continuous embeddings). In fact, if

$r<1\quad\text{and}\quad\\{t_{j}\\}_{j=0}^{\infty}\in\ell^{1}(\mathbf{N})\setminus\ell^{r}(\mathbf{N}),$

then

$f=\sum_{j=0}^{\infty}t_{j}\chi_{j+[0,1]}\in L^{1}(\mathbf{R})\setminus
W\\!\\!L^{1,r}(\mathbf{R}),$

which shows that $W\\!\\!L^{1,r}\neq L^{1}$. (Here $\chi_{E}$ is the
characteristic function of the set $E$.) This also shows that Theorem 0.2 is a
strict improvement of (0.8). We also observe that Theorems 0.1 and 0.2, and
(0.9) are not reachable by the classical results in [10, 30, 41, 42] when
$r<1$. In fact, these earlier investigations do not allow quasi-Banach spaces
like $W\\!\\!L^{1,r}$ which fails to be Banach spaces when $r<1$.

We also observe that the assertion in Theorems 0.1 and 0.2, and the identities
(0.7) and (0.9) fail to hold true when $p=\infty$. (See Remark 2.8.)

In Section 4 we consider factorizations for Schatten-von Neumann symbol
classes in pseudo-differential calculus with respect to convolutions, twisted
convolutions and symbolic products under compositions. For example, essential
achievements here is given in Theorem 4.13 which contains the following. Here
$s_{p}^{w}$ is the set of all symbols $\mathfrak{a}$ such that corresponding
Weyl operators $\operatorname{Op}^{w}(\mathfrak{a})$ is a Schatten-von Neumann
operator of order $p\in(0,\infty]$ on $L^{2}$, and
$\mathfrak{a}{\text{\footnotesize$\\#$}}\mathfrak{b}$ is the Weyl product of
$\mathfrak{a}$ and $\mathfrak{b}$ (see (4.19)).

######  Theorem 0.3.

Suppose $p\in(0,\sharp]$ and $r\in(0,1]$. Then

$\displaystyle W\\!\\!L^{1,r}*_{\sigma}s_{p}^{w}$
$\displaystyle=s_{\sharp}^{w}*_{\sigma}s_{p}^{w}=s_{\infty}^{w}*_{\sigma}s_{p}^{w}=s_{p}^{w}$
(0.10) and $\displaystyle W\\!\\!L^{1,r}{\text{\footnotesize$\\#$}}s_{p}^{w}$
$\displaystyle=s_{\sharp}^{w}{\text{\footnotesize$\\#$}}s_{p}^{w}=s_{\infty}^{w}{\text{\footnotesize$\\#$}}s_{p}^{w}=s_{p}^{w}.$
(0.11)

Using techniques of symplectic Fourier transform, it follows that the
identities (0.10) and (0.11) in the previous theorem are equivalent. Since

$W\\!\\!L^{1,r}\subsetneq L^{1}\subsetneq s_{\sharp}^{w}\subsetneq
s_{\infty}^{w}$ (0.12)

the identities in (0.10) and (0.11) at a first glance might be surprising. The
last equality $s_{\infty}^{w}{\text{\footnotesize$\\#$}}s_{p}^{w}=s_{p}^{w}$
in (0.11) is straight-forward, using Hölder’s inequality
$s_{\infty}^{w}{\text{\footnotesize$\\#$}}s_{p}^{w}\subseteq s_{p}^{w}$, and
that the identity $\mathfrak{a}_{0}(x,\xi)\equiv 1$ for $\\#$ belongs to
$s_{\infty}^{w}$.

The identities $s_{\sharp}^{w}{\text{\footnotesize$\\#$}}s_{p}^{w}=s_{p}^{w}$
and $L^{1}{\text{\footnotesize$\\#$}}s_{p}^{w}=s_{p}^{w}$ in (0.11), in the
case $p\geq 1$, follow e. g. from (0.12), the bounded approximate identity for
$L^{1}$ and Hewitt’s result [30, (2.5) Theorem]. Finally, the strongest
identity $W\\!\\!L^{1,r}{\text{\footnotesize$\\#$}}s_{p}^{w}=s_{p}^{w}$ in
(0.11) follow from the bounded approximate identity for $W\\!\\!L^{1,r}$ and
Theorem 0.4.

An essential ingredient of our investigations is an extension of Hewitt’s
factorization result in [30, (2.5) Theorem] to the quasi-Banach case, given by
the following.

######  Theorem 0.4.

Suppose $\mathcal{B}$ is a quasi-Banach algebra and $\mathcal{M}$ is a left
quasi-Banach $\mathcal{B}$ module. Also suppose $\mathcal{B}$ possess a
bounded left approximate identity for $(\mathcal{B},\mathcal{M})$. Then

$\mathcal{B}\cdot\mathcal{M}=\mathcal{M}.$

See Section 1 for descriptions of bounded left approximate identities.

A deep generalization of Theorem 0.4 was deduced by Ansari-Piri in [3, Theorem
4.1], who proved that [30, (2.5) Theorem] and Theorem 0.4 can be extended to
general _F_ -spaces. On the other hand, [30, (2.5) Theorem] is more general
compared to the factorization results by Salem in [44], Rudin in [41, 42] and
Cohen in [10]. In particular, Rudin’s identity

$L^{1}(G)*L^{1}(G)=L^{1}(G)$

when $G$ is any locally Euclidean Abelian group follows from [30, (2.5)
Theorem] (and thereby from Theorem 0.4 as well).

These factorization results have found immanence applications in science. To
give a flavor we note that in 1964, Varopoulos in [55] applies of Cohen’s
factorization theorem to prove that every positive linear functional on a
Banach∗-algebra with a bounded approximate identity is continuous. In 1976,
Green in [27] proves that every maximal left ideal in a Banach algebra with a
bounded right approximate identity is closed.

A more recent application concerns [7], where such factorization properties
are used to construct negative results on well-posedness for Hartree equation
in $L^{p}(\mathbf{R}^{d})\cap L^{2}(\mathbf{R}^{d})$ when $1<p\leq 2$. See [7,
Remark 3.6].

For Segal algebras, Feichtinger, Graham and Lakien showed in [21] that
factorizations can happen under rather weak assumptions. We also remark that
there are examples on Banach algebras having the factorization property, but
with absence of bounded approximate units (see e. g. [34, 57]).

We have included a proof of Theorem 0.4 in Appendix A, which follows the
framework of Hewitt’s proof in [30] in the case of Banach algebras.

The paper is organized as follows. In Section 1 we introduce needed notations
and recall basic facts for quasi-Banach spaces, algebras and modules, weight
functions used in time-frequency analysis, the Gelfand-Shilov space
$\Sigma_{1}(\mathbf{R}^{d})$ and its distribution space, and Wiener amalgam
and modulation spaces.

In Section 2 we apply Theorem 0.4 to prove factorization properties for quasi-
Banach Wiener amalgam under convolutions and twisted convolution. (See
Theorems 2.2 and 2.5.) In Section 3 we deduce factorization properties for a
large class of quasi-Banach modulation spaces under convolutions. (See Theorem
3.3.) The class of modulation spaces is explained in Subsection 1.1, and is
broader compared to what is usually considered. In Appendix B we show some
properties of such modulation spaces.

In Section 4 we first make a review of Schatten-von Neumann symbol classes in
calculi of pseudo-differential operators and introduce related items. Then we
apply the results from Section A to prove factorization properties for such
symbol classes under convolutions, twisted convolutions and symbol products
which are the operator compositions on the symbol side. (See Theorems
4.12–4.14.)

## Acknowledgement

We are grateful to professors H. G. Feichtinger and R. Fulsche for several
valuable comments and advices which lead to significantly improvements of the
content and style.

The first author is grateful to professor J. R. Patadia for introducing him
factorization problems during his M.Phil. thesis. The first author is also
grateful to professors K. Ross, S. Thangavelu and P.K. Ratnakumar for several
discussions and their encouragement on the topic.

The second author was supported by Vetenskapsrådet (Swedish Science Council),
within the project 2019-04890.

## Conflict of interest

On behalf of all authors, the corresponding author states that there is no
conflict of interest.

##  1\. Preliminaries

In this section we first review some facts on quasi-Banach algebras and
modules. Then we discuss weight functions. Thereafter we recall the definition
and basic properties on a suitable Gelfand-Shilov space and its distribution
space to have in background. Then we make a review of certain types of Wiener
amalgam spaces and consider a broad class of modulation spaces and discuss
general properties of such spaces.

###  1.1. Quasi-Banach algebras and modules

Let $\mathcal{B}$ be a vector space over $\mathbf{C}$. (We only consider
vector spaces over $\mathbf{C}$, but evidently our analysis also holds for
vector spaces over $\mathbf{R}$.) A quasi-norm of order $p\in(0,1]$, or a
$p$-norm on $\mathcal{B}$, is a map $f\mapsto\|f\|_{\mathcal{B}}$ from
$\mathcal{B}$ to $\mathbf{R}$ such that

$\displaystyle\|\phi+\psi\|_{\mathcal{B}}$ $\displaystyle\leq
2^{\frac{1}{p}-1}(\|\phi\|_{\mathcal{B}}+\|\psi\|_{\mathcal{B}}),$
$\displaystyle\quad\phi,\psi$ $\displaystyle\in\mathcal{B},$ (1.1)
$\displaystyle\|\alpha\phi\|_{\mathcal{B}}$
$\displaystyle=|\alpha|\,\|\phi\|_{\mathcal{B}},$ $\displaystyle\quad\phi$
$\displaystyle\in{\mathcal{B}},\ \alpha\in\mathbf{C},$ (1.2) and
$\displaystyle\|\phi\|_{\mathcal{B}}$ $\displaystyle>0,$
$\displaystyle\quad\phi$ $\displaystyle\in{\mathcal{B}}\setminus\\{0\\}.$
(1.3)

Evidently, $\|\phi\|_{\mathcal{B}}=0$, if and only if $\phi=0$, due to (1.2)
and (1.3).

The vector space $\mathcal{B}$ with $p$-norm $\|\,\cdot\,\|_{\mathcal{B}}$ is
called a quasi-Banach space of order $p$, if $\mathcal{B}$ is complete under
the vector space topology defined by $\|\,\cdot\,\|_{\mathcal{B}}$. By Aoki-
Rolewicz theorem it follows that for the quasi-Banach space $\mathcal{B}$ with
quasi-norm $\|\,\cdot\,\|_{\mathcal{B}}$ satisfying (1.1)– (1.3), there is an
equivalent quasi-norm of order $p$, which additionally satisfies

$\|\phi+\psi\|_{\mathcal{B}}^{p}\leq\|\phi\|_{\mathcal{B}}^{p}+\|\psi\|_{\mathcal{B}}^{p},\quad\phi,\psi\in\mathcal{B}.$
(1.4)

(See [1, 39].) From now on we assume that the quasi-norms to our quasi-Banach
spaces are chosen such that (1.1)–(1.4) hold true.

We have now the following definition.

######  Definition 1.1.

Let $p\in(0,1]$.

  1. (1)

The algebra $\mathcal{B}$ with $p$-norm $\|\,\cdot\,\|_{\mathcal{B}}$
satisfying (1.1)–(1.4), is called a _quasi-Banach algebra_ of order $p$, or a
$p$-Banach algebra, if $\mathcal{B}$ is a quasi-Banach space of order $p$
under $\|\,\cdot\,\|_{\mathcal{B}}$, and there is a constant $C>0$ such that

$\|\phi\psi\|_{\mathcal{B}}\leq
C\|\phi\|_{\mathcal{B}}\|\psi\|_{\mathcal{B}},\quad\phi,\psi\in\mathcal{B}\text{;}$
(1.5)

  2. (2)

if $\mathcal{B}$ is a quasi-Banach algebra of order $p$, then the left (right)
$\mathcal{B}$ module $\mathcal{M}$ with $p$-norm $\|\,\cdot\,\|_{\mathcal{M}}$
and multiplication $\cdot$ is called a _left (right) quasi-Banach
$\mathcal{B}$ module of order $p$_, or a $p$-Banach module over $\mathcal{B}$,
if $\mathcal{M}$ is a quasi-Banach space under $\|\,\cdot\,\|_{\mathcal{M}}$
of order $p$, and there is a constant $C>0$ such that

$\|\phi\cdot f\|_{\mathcal{M}}\leq
C\|\phi\|_{\mathcal{B}}\|f\|_{\mathcal{M}}\quad\big{(}\|f\cdot\phi\|_{\mathcal{M}}\leq
C\|\phi\|_{\mathcal{B}}\|f\|_{\mathcal{M}},\big{)},\quad\phi\in\mathcal{B},\
f\in\mathcal{M}.$ (1.6)

If $\mathcal{B}$ and $\mathcal{M}$ in Definition 1.1 are Banach spaces, then
$\mathcal{B}$ is called a Banach algebra and $\mathcal{M}$ is called a Banach
module over $\mathcal{B}$.

######  Remark 1.2.

Let $\mathcal{B}$ and $\mathcal{M}$ be as in Definition 1.1, and let
$f\in\mathcal{M}$. Then

$\mathcal{M}_{f}\equiv\overline{\\{\,\phi\cdot
f\,;\,\phi\in\mathcal{B}\,\\}}\subseteq\mathcal{M}$ (1.7)

is a quasi-Banach $\mathcal{B}$ submodule to $\mathcal{M}$ of order $p$. Here
$\overline{\Omega}$ denotes the closure of $\Omega\subseteq\mathcal{M}$ in
$\mathcal{M}$.

By replacing the quasi-norms $\|\,\cdot\,\|_{\mathcal{B}}$ and
$\|\,\cdot\,\|_{\mathcal{M}}$ in Definition 1.1 with
$c\cdot\|\,\cdot\,\|_{\mathcal{B}}$ and $c\cdot\|\,\cdot\,\|_{\mathcal{M}}$,
respectively, for some $c>0$, it follows that (1.5) and (1.6) hold true with
$C=1$. That is, we may reduce ourselves to the case when (1.5) and (1.6) take
the forms

$\displaystyle\|\phi\psi\|_{\mathcal{B}}$
$\displaystyle\leq\|\phi\|_{\mathcal{B}}\|\psi\|_{\mathcal{B}},$
$\displaystyle\quad\phi,\psi$ $\displaystyle\in\mathcal{B}$ (1.5)′ and
$\displaystyle\|\phi\cdot f\|_{\mathcal{M}}$
$\displaystyle\leq\|\phi\|_{\mathcal{B}}\|f\|_{\mathcal{M}},$
$\displaystyle\quad\phi$ $\displaystyle\in\mathcal{B},\ f\in\mathcal{M}.$
(1.6)′

We also have the following definition.

######  Definition 1.3.

Let $\mathcal{B}$ be a quasi-Banach algebra of order $p\in(0,1]$ and
$\mathcal{M}$ be a left quasi-Banach $\mathcal{B}$ module of order $p$.

  1. (1)

$\mathcal{B}$ is said to _possess a bounded left approximate identity or unit_
of order $r>0$, if for every $\varepsilon>0$ and finite set
$\\{\psi_{1},\dots,\psi_{n}\\}\subseteq\mathcal{B}$, there is an
$\phi_{\varepsilon}\in\mathcal{B}$ such that

$\|\phi_{\varepsilon}\psi_{j}-\psi_{j}\|_{\mathcal{B}}<\varepsilon,\
j=1,\dots,n,\quad\text{and}\quad\|\phi_{\varepsilon}\|_{\mathcal{B}}\leq r.$
(1.8)

  2. (2)

$\mathcal{B}$ is said to _possess a bounded left approximate identity or unit
for $\mathcal{M}$_ of order $r>0$, if for every $\varepsilon>0$ and
$f\in\mathcal{M}$, there is an $\phi_{\varepsilon}\in\mathcal{B}$ such that

$\|\phi_{\varepsilon}\cdot
f-f\|_{\mathcal{M}}<\varepsilon\quad\text{and}\quad\|\phi_{\varepsilon}\|_{\mathcal{B}}\leq
r.$ (1.9)

  3. (3)

$\mathcal{B}$ is said to _possess a bounded left approximate identity or unit
for $(\mathcal{B},\mathcal{M})$_ of order $r>0$, if for every $\varepsilon>0$,
finite set $\\{\psi_{1},\dots,\psi_{n}\\}\subseteq\mathcal{B}$ and
$f\in\mathcal{M}$, there is an $\phi_{\varepsilon}\in\mathcal{B}$ such that
(1.8) and (1.9) hold true.

######  Remark 1.4.

We observe that a quasi-Banach algebra $\mathcal{B}$ has a bounded left
approximate identity, if and only if there exist a net
$\\{\phi_{\lambda}\\}_{\lambda\in\Lambda}\subseteq\mathcal{B}$ and a constant
$r>0$ such that

$\lim_{\lambda\in\Lambda}\phi_{\lambda}\psi=\psi,\quad\psi\in\mathcal{B}$

and $\|\phi_{\lambda}\|_{\mathcal{B}}\leq r$ for every $\lambda\in\Lambda$.
(See [13] for details.)

We have now the following factorization result, which is an extension of [30,
(2.5) Theorem], but a special case of [3, Theorem 4.1].

######  Theorem 1.5.

Suppose that $\mathcal{B}$ is a quasi-Banach algebra of order $p\in(0,1]$ and
that $\mathcal{M}$ is a left quasi-Banach $\mathcal{B}$ module of order $p$.
Also suppose that $\mathcal{B}$ possess a bounded left approximate identity
for $(\mathcal{B},\mathcal{M})$ of order $r>0$. Let $f\in\mathcal{M}$,
$\mathcal{M}_{f}$ be the same as in (1.7), set

$r_{0}=\frac{r}{((2r+1)^{p}-(2r)^{p})^{\frac{1}{p}}},$

and let $\varepsilon>0$ be arbitrary. Then there exist elements $\psi\in
B_{r_{0},\mathcal{B}}$ and $g\in\mathcal{M}_{f}$ such that the following is
true:

  1. (1)

$f=\psi\cdot g$;

  2. (2)

$\|f-g\|_{\mathcal{M}}<\varepsilon$.

For conveniency to the reader, we give a proof of Theorem 0.4 in Appendix A,
following the framework of Hewitt’s proof in [30] in the restricted case of
Banach algebras.

By Theorem 1.5 it follows that a quasi-Banach algebra having either a bounded
left or right approximate identity is a factorization algebra (cf. Definition
1.10). However, the converse may not be true, i. e., a quasi-Banach algebra
which is a factorization algebra, might not contain any bounded left or right
approximate identity (see e. g. [34, 57]). On the other hand, under suitable
factorization assumptions, the converse is true, as in the following
proposition.

######  Proposition 1.6.

Let $\mathcal{B}$ be a quasi-Banach algebra and suppose there is a constant
$C>0$ such that for every $\phi\in\mathcal{B}$ and $\delta>0$, there are
elements $\phi_{0},\psi\in\mathcal{B}$ such that

$\phi=\phi_{0}\psi,\quad\|\psi\|_{\mathcal{B}}\leq
C,\quad\text{and}\quad\|f-g\|_{\mathcal{B}}<\delta.$

Then $\mathcal{B}$ has a bounded left approximate identity.

A proof of Proposition 1.6, in the restricted case when $\mathcal{B}$ is a
Banach algebra, can be found in e. g. [13]. For general $\mathcal{B}$, the
result follows by similar arguments as in [13]. We omit the details.

######  Example 1.7.

If $p\in[1,\infty)$, then $\mathcal{B}=L^{1}(\mathbf{R}^{d})$ is a Banach
algebra under convolution, and $\mathcal{M}=L^{p}(\mathbf{R}^{d})$ is a Banach
module to $L^{1}(\mathbf{R}^{d})$ under convolution. A classical example on
bounded approximate identity for such algebras and modules are mollifiers

$\\{\phi_{\varepsilon}\\}_{\varepsilon>0},\quad\text{where}\quad\phi_{\varepsilon}(x)=\varepsilon^{-d}\phi({\textstyle{\frac{x}{\varepsilon}}}),\
\phi\in L^{1}(\mathbf{R}^{d}),\ \int_{\mathbf{R}^{d}}\phi(x)\,dx=1.$ (1.10)

More generally, let $r\in(0,1]$ and $p\in[1,\infty)$, $q\in[r,\infty)$, and
$\mathcal{B}=W\\!\\!L^{1,r}(\mathbf{R}^{d})$ and
$\mathcal{M}=W\\!\\!L^{p,q}(\mathbf{R}^{d})$ be as in Theorem 0.1 in the
introduction. Then $\mathcal{B}$ is a quasi-Banach algebra and $\mathcal{M}$
is a quasi-Banach module over $\mathcal{B}$ under convolution. (See
Proposition 2.1 in Section 2.) Again it follows that the mollifiers in (1.10)
are bounded left identities, provided, more restrictive, $\phi\in
W\\!\\!L^{1,r}(\mathbf{R}^{d})$. Here we also observe that $\mathcal{B}$ and
$\mathcal{M}$ are _not_ Banach spaces when $r<1$ and $\min(p,q)<1$.

We also need to extend our quasi-Banach algebras where a unit element is
included.

######  Definition 1.8.

Let $\mathcal{B}$ be a quasi-Banach algebra of order $p\in(0,1]$, and let
$\mathcal{M}$ be a left quasi-Banach $\mathcal{B}$ module of order $p$.

  1. (1)

if $\mathcal{B}$ is unital, then let $\mathcal{B}_{E}=\mathcal{B}$, and let
$\varrho$ denote the unit element in $\mathcal{B}_{E}=\mathcal{B}$.

  2. (2)

if $\mathcal{B}$ is not unital and $\varrho\notin\mathcal{B}$, then let
$\mathcal{B}_{E}\equiv(\mathbf{C}\varrho)\oplus\mathcal{B}$ be the algebra,
equipped with the operations

$(s\varrho+\phi)(t\varrho+\psi)=st\varrho+(\phi\psi+t\phi+s\psi),\qquad
s,t\in\mathbf{C},\ \phi,\psi\in\mathcal{B},$

and with quasi-norm

$\|t\varrho+\phi\|_{\mathcal{B}_{E}}\equiv\left(|t|^{p}+\|\phi\|_{\mathcal{B}}^{p}\right)^{\frac{1}{p}},\qquad\phi\in\mathcal{B},\
t\in\mathbf{C}.$ (1.11)

  3. (3)

$\mathcal{B}_{E}$ acts on $\mathcal{M}$ by the formula

$(t\varrho+\phi)\cdot f=tf+\phi\cdot f,\qquad t\in\mathbf{C},\
\phi\in\mathcal{B},\ f\in\mathcal{M}.$

It is evident that $\mathcal{B}_{E}$ in Definition 1.8 is a unital quasi-
Banach algebra of order $p$ containing $\mathcal{B}$, and with identity
element $\varrho$. For the quasi-norm in (1.11) for $\mathcal{B}_{E}$
restricted to $\mathcal{B}$ one has

$\|\phi\|_{\mathcal{B}_{E}}=\|\phi\|_{\mathcal{B}}\quad\text{when}\quad\phi\in\mathcal{B}.$

######  Example 1.9.

The space $\mathcal{B}=L^{1}(\mathbf{R}^{d})$ is a Banach algebra under
convolution. Since $L^{1}(\mathbf{R}^{d})$ does not contain any identity
element, it follows that

$\mathcal{B}_{E}=(\mathbf{C}\varrho)\oplus\mathcal{B}$

where $\varrho=\delta_{0}$ is the Dirac’s delta function. In particular,
$\mathcal{B}_{E}\neq\mathcal{B}$ in this case.

Let instead $\mathcal{B}$ be the Banach space $\mathcal{M}(\mathbf{R}^{d})$ of
(complex-valued) measures on $\mathbf{R}^{d}$ with bounded mass. Recall that
the convolution $\mu*\nu\in\mathcal{M}(\mathbf{R}^{d})$ of
$\mu,\nu\in\mathcal{M}(\mathbf{R}^{d})$ is defined by

$\langle\mu*\nu,\varphi\rangle=\iint_{\mathbf{R}^{2d}}\varphi(x+y)\,d\mu(x)d\nu(y),\quad\varphi\in
C_{0}(\mathbf{R}^{d}).$

Then

  * •

$(\mathcal{M}(\mathbf{R}^{d}),*)$ is a commutative Banach superalgebra of the
Banach algebra $(L^{1}(\mathbf{R}^{d}),*)$, in view of Fubini’s theorem. Here
we identify any $f\in L^{1}(\mathbf{R}^{d})$ with the measure
$\mu_{f}(x)=f(x)\,dx\in\mathcal{M}(\mathbf{R}^{d})$, and observe that
$\|\mu_{f}\|_{{\mathcal{M}}}=\|f\|_{L^{1}}$.

  * •

$\mathcal{B}_{E}=\mathcal{B}$ in this case, because
$\varrho=\delta_{0}\in\mathcal{M}(\mathbf{R}^{d})$.

Our investigations concerns factorization properties for certain algebras in
the following sense.

######  Definition 1.10.

Let $\mathscr{A}$ be an algebra and $\mathscr{M}$ be a left (right) module
over $\mathscr{A}$.

  1. (1)

$\mathscr{A}$ is called a _factorization algebra_ or _possess factorization
property_ if for every $\phi\in\mathscr{A}$, there are
$\phi_{1},\phi_{2}\in\mathscr{A}$ such that $\phi=\phi_{1}\phi_{2}$.

  2. (2)

$\mathscr{M}$ is called a _left (right) factorization module_ or _possess
factorization property_ if for every $f\in\mathscr{M}$, there are
$\phi_{0}\in\mathscr{A}$ and $f_{0}\in\mathscr{M}$ such that $f=\phi_{0}\cdot
f_{0}$ ($f=f_{0}\cdot\phi_{0}$).

In some examples later on, it is convenient to have the following abstract
result in hands.

######  Proposition 1.11.

Let $\mathscr{A}$ be an algebra, $\mathscr{A}_{0}$ a factorization subalgebra
to $\mathscr{A}$, $G$ be an ordered semi-group with ordering $\prec$ and let
$\\{\mathscr{A}_{g}\\}_{g\in G}$ be a family of subalgebras to $\mathscr{A}$
such that the following is true:

  * •

if $g_{1}\prec g_{2}$, then $\mathscr{A}_{g_{2}}\subseteq\mathscr{A}_{g_{1}}$;

  * •

if $g\in G$ and $m,n\in\mathbf{N}$ satisfy $m<n$, then $g^{m}\prec g^{n}$;

  * •

if $g_{0}\in G$ is fixed, then

$\bigcap_{n\in\mathbf{Z}_{+}}\mathscr{A}_{g_{0}^{n}}=\bigcap_{g\in
G}\mathscr{A}_{g}.$ (1.12)

Then the following conditions are equivalent:

  1. (1)

$\mathscr{A}_{0}\subseteq\mathscr{A}_{g_{0}}$ for some $g_{0}\in G$;

  2. (2)

$\mathscr{A}_{0}\subseteq\mathscr{A}_{g}$ for every $g\in G$.

###### Proof.

Evidently, (2) implies (1). Therefore assume that (1) holds, and let
$\phi\in\mathscr{A}$ be arbitrary. Also let $n\in\mathbf{Z}_{+}$ be arbitrary.
Since $\mathscr{A}$ is a factorization algebra, there are
$\phi_{1},\dots,\phi_{n}\in\mathscr{A}_{0}\subseteq\mathscr{A}_{g_{0}}$ such
that

$\phi=\phi_{1}\cdots\phi_{n}\in\mathscr{A}_{g_{0}}\cdots\mathscr{A}_{g_{0}}=\mathscr{A}_{g_{0}^{n}}.$

Hence $\phi\in\mathscr{A}_{g_{0}^{n}}$ for every $n\in\mathbf{Z}_{+}$. By
(1.12) it follows that $\phi\in\mathscr{A}_{g}$ for every $g\in G$. Since
$\phi\in\mathscr{A}_{0}$ was arbitrarily chosen it follows that
$\mathscr{A}_{0}\subseteq\mathscr{A}_{g}$ for every $g\in G$, and the result
follows. ∎

###  1.2. Weight functions

A _weight_ or _weight function_ $\omega$ on $\mathbf{R}^{d}$ is a positive
function such that $\omega,1/\omega\in L^{\infty}_{loc}(\mathbf{R}^{d})$. Let
$\omega$ and $v$ be weights on $\mathbf{R}^{d}$. Then $\omega$ is called _$v$
-moderate_ or _moderate_ , if

$\omega(x+y)\lesssim\omega(x)v(y),\quad x,y\in\mathbf{R}^{d}.$ (1.13)

Here $f(\theta)\lesssim g(\theta)$ means that $f(\theta)\leq cg(\theta)$ for
some constant $c>0$ which is independent of $\theta$ in the domain of $f$ and
$g$. We also write $f(\theta)\asymp g(\theta)$ when $f(\theta)\lesssim
g(\theta)\lesssim f(\theta)$.

The function $v$ is called _submultiplicative_ , if it is even and (1.13)
holds for $\omega=v$. We notice that (1.13) implies that if $v$ is
submultiplicative on $\mathbf{R}^{d}$, then there is a constant $c>0$ such
that $v(x)\geq c$ when $x\in\mathbf{R}^{d}$.

We let $\mathscr{P}_{E}(\mathbf{R}^{d})$ be the set of all moderate weights on
$\mathbf{R}^{d}$. We recall that if
$\omega\in\mathscr{P}_{E}(\mathbf{R}^{d})$, then $\omega$ is moderated by
$v(x)=e^{r|x|}$, provided the constant $r\geq 0$ is chosen large enough. A
combination of this fact with (1.13) implies that

$e^{-r|x|}\lesssim\omega(x)\lesssim
e^{r|x|},\quad\omega\in\mathscr{P}_{E}(\mathbf{R}^{d}),$ (1.14)

where $r$ depends on $\omega$. (See [28].)

###  1.3. A Gelfand-Shilov space

In several parts of our investigations it is suitable to perform the
discussions in the framework of the Gelfand-Shilov space
$\Sigma_{1}(\mathbf{R}^{d})$ of Beurling type, which consists of all $f\in
C^{\infty}(\mathbf{R}^{d})$ such that

$\sup_{\alpha,\beta\in{\mathbf{N}}^{d}}\left(\frac{\|x^{\alpha}D^{\beta}f\|_{L^{\infty}}}{h^{|\alpha+\beta|}\alpha!\beta!}\right)$
(1.15)

is finite for every $h>0$ (cf. [35]). It follows that
$\Sigma_{1}(\mathbf{R}^{d})$ is a Fréchet space under the semi-norms in
(1.15). The (strong) dual of $\Sigma_{1}(\mathbf{R}^{d})$, i. e. the (Gelfand-
Shilov) distribution space of $\Sigma_{1}(\mathbf{R}^{d})$, is denoted by
$\Sigma_{1}^{\prime}(\mathbf{R}^{d})$. It follows that the duality between
$\Sigma_{1}(\mathbf{R}^{d})$ and $\Sigma_{1}^{\prime}(\mathbf{R}^{d})$ is
given by a unique extension of the $L^{2}$ scalar product
$(\,\cdot\,,\,\cdot\,)_{L^{2}}$ on
$\Sigma_{1}(\mathbf{R}^{d})\times\Sigma_{1}(\mathbf{R}^{d})$. We have

$\Sigma_{1}(\mathbf{R}^{d})\subsetneq\mathscr{S}(\mathbf{R}^{d})\subsetneq
L^{2}(\mathbf{R}^{d})\subsetneq\mathscr{S}^{\prime}(\mathbf{R}^{d})\subsetneq\Sigma_{1}^{\prime}(\mathbf{R}^{d})$

with continuous and dense embeddings.

The spaces $\Sigma_{1}(\mathbf{R}^{d})$ and
$\Sigma_{1}^{\prime}(\mathbf{R}^{d})$ possess several convenient properties
and characterizations. If $\mathscr{F}$ denotes the Fourier transform

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

when $f\in L^{1}(\mathbf{R}^{d})$, then $\mathscr{F}$ restricts to
homeomorphisms on $\mathscr{S}(\mathbf{R}^{d})$ and on
$\Sigma_{1}(\mathbf{R}^{d})$. The definition of $\mathscr{F}$ on
$\Sigma_{1}(\mathbf{R}^{d})$ extends uniquely to homeomorphisms on
$\mathscr{S}^{\prime}(\mathbf{R}^{d})$ and on
$\Sigma_{1}^{\prime}(\mathbf{R}^{d})$, and to an isometric bijection on
$L^{2}(\mathbf{R}^{d})$. The inverse of $\mathscr{F}$ is given by

$\mathscr{F}^{-1}=U\circ\mathscr{F}=\mathscr{F}\circ
U,\quad\text{when}\quad(Uf)(x)=f(-x),\
f\in\Sigma_{1}^{\prime}(\mathbf{R}^{d}).$

In particular, the Fourier’s inversion formula is given by

$f(x)=(\mathscr{F}^{-1}\widehat{f})(x)=(2\pi)^{-\frac{d}{2}}\int_{\mathbf{R}^{d}}\widehat{f}(\xi)e^{i\langle
x,\xi\rangle}\,d\xi,$

when $\widehat{f}\in L^{1}(\mathbf{R}^{d})$.

The following result shows that $\Sigma_{1}(\mathbf{R}^{d})$ and
$\Sigma_{1}^{\prime}(\mathbf{R}^{d})$ may in convenient ways be characterized
by growth and decay conditions on their Fourier transforms and their short-
time Fourier transform. Here recall that the short-time Fourier transform of a
distribution $f\in\Sigma_{1}^{\prime}(\mathbf{R}^{d})$ with respect to the
window function $\phi\in\Sigma_{1}(\mathbf{R}^{d})\setminus 0$ is defined as

$\displaystyle V_{\phi}f(x,\xi)$
$\displaystyle\equiv\mathscr{F}(f\cdot\overline{\phi(\,\cdot\,-x)})(\xi),$
$\displaystyle\quad x,\xi$ $\displaystyle\in\mathbf{R}^{d}.$ We note that for
$f\in L^{p}(\mathbf{R}^{d})$ for some $p\in[1,\infty]$, then $\displaystyle
V_{\phi}f(x,\xi)$
$\displaystyle\equiv(2\pi)^{-\frac{d}{2}}\int_{\mathbf{R}^{d}}f(y)\overline{\phi(y-x)}e^{-i\langle
y,\xi\rangle}\,d\xi,$ $\displaystyle\quad x,\xi$
$\displaystyle\in\mathbf{R}^{d}.$

In some parts of our exposition it is more convenient to use the transform

$\mathcal{T}_{\phi}f(x,\xi)\equiv\mathscr{F}(f(\,\cdot\,+x)\cdot\overline{\phi})(\xi)=e^{i\langle
x,\xi\rangle}V_{\phi}f(x,\xi),\quad x,\xi\in\mathbf{R}^{d},$

instead of the short-time Fourier transform, because of the simple convolution
relation

$(\mathcal{T}_{\phi}(f*g))(x,\xi)=\big{(}(\mathcal{T}_{\phi}f)(\,\cdot\,,\xi)*g\big{)}(x).$
(1.16)

######  Proposition 1.12.

Let $\phi\in\Sigma_{1}(\mathbf{R}^{d})\setminus 0$ be fixed and let
$f\in\Sigma_{1}^{\prime}(\mathbf{R}^{d})$. Then the following is true:

  1. (1)

it holds

$|V_{\phi}f(x,\xi)|=|\mathcal{T}_{\phi}f(x,\xi)|\lesssim e^{r(|x|+|\xi|)}$

for some $r>0$;

  2. (2)

$f\in\Sigma_{1}(\mathbf{R}^{d})$, if and only if

$|V_{\phi}f(x,\xi)|=|\mathcal{T}_{\phi}f(x,\xi)|\lesssim e^{-r(|x|+|\xi|)}$

for every $r>0$;

  3. (3)

$f\in\Sigma_{1}(\mathbf{R}^{d})$, if and only if

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

for every $r>0$.

We refer to [48] for the proof of (1), [29] for the proof of (2) and [9] for
the proof of (3) in Proposition 1.12.

###  1.4. Wiener amalgam and modulation spaces

Let $p\in[1,\infty]$, $q\in(0,\infty]$ and
$\omega\in\mathscr{P}_{E}(\mathbf{R}^{d})$. Then the Wiener amalgam space
$W\\!\\!L^{p,q}_{(\omega)}(\mathbf{R}^{d})$ consists of all measurable
functions $f$ on $\mathbf{R}^{d}$ such that
$\|f\|_{W\\!\\!L^{p,q}_{(\omega)}}$ is finite, where

$\displaystyle\|f\|_{W\\!\\!L^{p,q}_{(\omega)}}$
$\displaystyle\equiv\|a_{f}\|_{\ell^{q}_{(\omega)}(\mathbf{Z}^{d})},$ (1.17)
$\displaystyle a_{f}(k)$ $\displaystyle=\|f\|_{L^{p}(k+Q)},\quad
k\in\mathbf{Z}^{d},\ Q=[0,1]^{d}.$

The set $W\\!\\!L^{p,q}_{(\omega)}(\mathbf{R}^{d})$ with quasi-norm (1.17) is
a quasi-Banach space of order $\min(q,1)$. In particular,
$W\\!\\!L^{p,q}_{(\omega)}(\mathbf{R}^{d})$ is a Banach space when $q\geq 1$.
We observe that if $p,q<\infty$, then $C_{0}^{\infty}(\mathbf{R}^{d})$ and
$\Sigma_{1}(\mathbf{R}^{d})$ are dense in
$W\\!\\!L^{p,q}_{(\omega)}(\mathbf{R}^{d})$. For convenience we set
$W\\!\\!L^{p,q}=W\\!\\!L^{p,q}_{(\omega)}$ when $\omega(x)=1$ for every
$x\in\mathbf{R}^{d}$. We also recall that
$W\\!\\!L^{p,q}_{(\omega)}(\mathbf{R}^{d})$ is often denoted by

$W(L^{p}(\mathbf{R}^{d}),L^{q}_{(\omega)}(\mathbf{R}^{d}))=W(L^{p}(\mathbf{R}^{d}),\ell^{q}_{(\omega)}(\mathbf{Z}^{d})),$

in the literature. (See e. g. [18, 22].) We observe that
$W\\!\\!L^{p,p}_{(\omega)}(\mathbf{R}^{d})$ is equal to
$L^{p}_{(\omega)}(\mathbf{R}^{d})$, the set of all measurable functions $f$ on
$\mathbf{R}^{d}$ such that
$\|f\|_{L^{p}_{(\omega)}}\equiv\|f\cdot\omega\|_{L^{p}}$ is finite. Hence the
Wiener amalgam spaces extend the notion on Lebesgue spaces.

Next we recall the definition of _classical modulation spaces_ (see e. g. [16,
26]). Let $p,q\in(0,\infty]$ be fixed, and set

$\|F\|_{L^{p,q}(\mathbf{R}^{2d})}\equiv\|f_{p}\|_{L^{q}(\mathbf{R}^{d})},\qquad
f_{p}(\xi)\equiv\|F(\,\cdot\,,\xi)\|_{L^{p}(\mathbf{R}^{d})},$

when $F$ is measurable on $\mathbf{R}^{2d}$. Then $L^{p,q}(\mathbf{R}^{2d})$
is the quasi-Banach space with quasi-norm $\|\,\cdot\,\|_{L^{p,q}}$ which
consists of all measurable functions $F$ on $\mathbf{R}^{2d}$ such that
$\|F\|_{L^{p,q}(\mathbf{R}^{2d})}$ is finite. We observe that if $p,q\geq 1$,
then $L^{p,q}(\mathbf{R}^{2d})$ is a Banach space and
$\|\,\cdot\,\|_{L^{p,q}}$ is a norm.

######  Definition 1.13.

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

  1. (1)

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

$\|f\|_{M^{p,q}_{(\omega)}}\equiv\|\mathcal{T}_{\phi}f\cdot\omega\|_{L^{p,q}}\asymp\|V_{\phi}f\cdot\omega\|_{L^{p,q}}$
(1.18)

is finite.

  2. (2)

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

$\|f\|_{W^{p,q}_{(\omega)}}\equiv\|G\|_{L^{q,p}}<\infty,\quad\text{where}\quad
G(\xi,x)=V_{\phi}f(x,\xi)\cdot\omega(x,\xi).$ (1.19)

For convenience we set
$M^{p}_{(\omega)}=M^{p,p}_{(\omega)}=W^{p,p}_{(\omega)}$. We also set

$M^{p,q}_{(\omega)}=M^{p,q},\quad
W^{p,q}_{(\omega)}=W^{p,q}\quad\text{and}\quad
M^{p}_{(\omega)}=M^{p},\quad\text{when}\quad\omega=1.$

We observe that the modulation spaces in Definition 1.13 (2) is certain types
of Wiener amalgam spaces (see e. g. [16]). The modulation spaces in Definition
1.13 (1) were introduced in [18] by Feichtinger, and is often refer as
_classical modulation spaces_.

There are several extensions of classical modulation spaces (see e. g. [5, 19,
22, 36, 38, 48, 53, 54]). We shall consider a broader family of modulation
spaces, parameterized by quasi-Banach function spaces given in the following
definition.

######  Definition 1.14.

Let $\mathscr{B}$ be a quasi-Banach space of measurable functions on
$\mathbf{R}^{d}$ of order $r\in(0,1]$ containing $\Sigma_{1}(\mathbf{R}^{d})$
with continuous embedding, and let $v_{0}\in\mathscr{P}_{E}(\mathbf{R}^{d})$.
Then $\mathscr{B}$ is called a _translation invariant quasi-Banach function
space on $\mathbf{R}^{d}$_, or _invariant QBF space on $\mathbf{R}^{d}$_,
(with respect to $v_{0}$ of order $r$) if there is a constant $C$ such that
the following is true:

  1. (1)

if $x\in\mathbf{R}^{d}$ and $f\in\mathscr{B}$, then
$f(\,\cdot\,-x)\in\mathscr{B}$, and

$\|f(\,\cdot\,-x)\|_{\mathscr{B}}\leq Cv_{0}(x)\|f\|_{\mathscr{B}}\text{;}$
(1.20)

  2. (2)

if $f$ and $g$ are measurable on $\mathbf{R}^{d}$ which satisfy
$g\in\mathscr{B}$ and $|f|\leq|g|$, then $f\in\mathscr{B}$ and

$\|f\|_{\mathscr{B}}\leq C\|g\|_{\mathscr{B}}\text{.}$

An invariant QBF space on $\mathbf{R}^{d}$ which is a Banach space is called a
_translation invariant Banach function space on $\mathbf{R}^{d}$_ or
_invariant BF space on $\mathbf{R}^{d}$_.

######  Definition 1.15.

Let $\mathscr{B}$ be an invariant QBF space on $\mathbf{R}^{2d}$ of order
$r\in(0,1]$ and let $\phi\in\Sigma_{1}(\mathbf{R}^{d})\setminus 0$.

  1. (1)

The modulation space $M(\omega,\mathscr{B})$ consists of all
$f\in\Sigma_{1}^{\prime}(\mathbf{R}^{d})$ such that

$\|f\|_{M(\omega,\mathscr{B})}\equiv\|\mathcal{T}_{\phi}f\cdot\omega\|_{\mathscr{B}}\asymp\|V_{\phi}f\cdot\omega\|_{\mathscr{B}}$
(1.21)

is finite.

  2. (2)

The modulation space $M_{0}(\omega,\mathscr{B})$ is the completion of
$\Sigma_{1}(\mathbf{R}^{d})$ under the norm
$\|\,\cdot\,\|_{M(\omega,\mathscr{B})}$.

  3. (3)

The modulation spaces $M(\omega,\mathscr{B})$ and $M_{0}(\omega,\mathscr{B})$
are called _normal_ if
$M(\omega,\mathscr{B})\hookrightarrow\Sigma_{1}^{\prime}(\mathbf{R}^{d})$ and
$M_{0}(\omega,\mathscr{B})\hookrightarrow\Sigma_{1}^{\prime}(\mathbf{R}^{d})$,
respectively.

######  Remark 1.16.

Let $p\in(0,\infty]$ be fixed, $v_{r}(x,\xi)=e^{r(|x|+|\xi|)}$ when $r>0$ is
fixed and $x,\xi\in\mathbf{R}^{d}$. Then $\Sigma_{1}^{\prime}(\mathbf{R}^{d})$
is the inductive limit of $M^{p}_{(1/v_{r})}(\mathbf{R}^{d})$ with respect to
$r>0$ (see e. g. [48]). This implies that $M(\omega,\mathscr{B})$ is normal,
if and only if

$M(\omega,\mathscr{B})\hookrightarrow M^{p}_{(1/v_{r})}(\mathbf{R}^{d}),$
(1.22)

for some $r>0$, and then

$\|f\|_{M^{p}_{(1/v_{r})}}\lesssim\|f\|_{M(\omega,\mathscr{B})},\qquad
f\in\Sigma_{1}^{\prime}(\mathbf{R}^{d}).$ (1.23)

We also observe that if $\mathscr{B}$ is as in Definition 1.15, then

$\Sigma_{1}(\mathbf{R}^{d})\hookrightarrow M(\omega,\mathscr{B}).$ (1.24)

This follows by a straight-forward combination of (1.14), Proposition 1.12 and
(1.20). Since $\Sigma_{1}(\mathbf{R}^{d})$ is the projective limit of
$M^{p}_{(v_{r})}(\mathbf{R}^{d})$ with respect to $r>0$, it follows that
$M(\omega,\mathscr{B})$ is normal, if and only if

$M^{p}_{(v_{r})}(\mathbf{R}^{d})\hookrightarrow
M(\omega,\mathscr{B})\hookrightarrow M^{p}_{(1/v_{r})}(\mathbf{R}^{d}),$
(1.22)′

for some $r>0$, and then

$\|f\|_{M^{p}_{(1/v_{r})}}\lesssim\|f\|_{M(\omega,\mathscr{B})}\lesssim\|f\|_{M^{p}_{(v_{r})}},\qquad
f\in\Sigma_{1}^{\prime}(\mathbf{R}^{d}).$ (1.23)′

######  Remark 1.17.

Let $\omega\in\mathscr{P}_{E}(\mathbf{R}^{2d})$ and $\mathscr{B}$ be an
invariant QBF spaces on $\mathbf{R}^{2d}$. By the previous remark it follows
that a sufficient condition for $M(\omega,\mathscr{B})$ to be normal is that

$\mathscr{B}\hookrightarrow L^{p}_{(1/v)}(\mathbf{R}^{2d}),$

for some $p\in(0,\infty]$ and some submultiplicative
$v\in\mathscr{P}_{E}(\mathbf{R}^{2d})$.

In the following proposition we list some basic properties for modulation
spaces.

######  Proposition 1.18.

Let $\omega\in\mathscr{P}_{E}(\mathbf{R}^{2d})$, and let $\mathscr{B}$ be an
invariant QBF space on $\mathbf{R}^{2d}$ of order $p\in(0,1]$. Then the
following is true:

  1. (1)

$M(\omega,\mathscr{B})$ and $M_{0}(\omega,\mathscr{B})$ are independent of the
choices of $\phi\in\Sigma_{1}(\mathbf{R}^{d})\setminus 0$ in (1.21), and
different choices of $\phi$ give rise to equivalent quasi-norms;

  2. (2)

if $M(\omega,\mathscr{B})$ is normal, then $M_{0}(\omega,\mathscr{B})$ is
normal, and $M(\omega,\mathscr{B})$ and $M_{0}(\omega,\mathscr{B})$ are quasi-
Banach spaces of order $p$.

A proof of Proposition 1.18 is available for certain choices of $\mathscr{B}$
and window functions $\omega$ and $v_{0}$ in e. g. [16, 22, 26, 28, 37, 38].
For general choices of $\mathscr{B}$ and window functions, we may argue in
similar ways. In order to be self-contained we present a proof of Proposition
1.18 in Appendix B.

The next result shows the role of $\omega$ concerning the magnitude of
$M(\omega,\mathscr{B})$.

######  Proposition 1.19.

Let $\omega_{1},\omega_{2}\in\mathscr{P}_{E}(\mathbf{R}^{2d})$, $\mathscr{B}$
be an invariant QBF-space on $\mathbf{R}^{2d}$ with respect to
$v\in\mathscr{P}_{E}(\mathbf{R}^{2d})$. Then the following is true:

  1. (1)

if ${\omega_{2}}\lesssim{\omega_{1}}$, then

$\displaystyle M_{0}(\omega_{1},\mathscr{B})\subseteq
M_{0}(\omega_{2},\mathscr{B}),\quad\text{and}\quad M(\omega_{1},\mathscr{B})$
$\displaystyle\subseteq M(\omega_{2},\mathscr{B}),$

and the injections

$\displaystyle i\\!:M_{0}(\omega_{1},\mathscr{B})\to
M_{0}(\omega_{2},\mathscr{B})\quad\text{and}\quad
i\\!:M(\omega_{1},\mathscr{B})\to M(\omega_{2},\mathscr{B})$ (1.25)

are continuous;

  2. (2)

if in addition $v$ is bounded, then the following conditions are equivalent:

     * •

at least one of the mappings in (1.25) is a continuous injection;

     * •

all the mappings in (1.25) are continuous injections;

     * •

${\omega_{2}}\lesssim{\omega_{1}}$.

Proposition 1.19 slightly generalizes parts of [36, Theorem 3.7]. In order to
be self-contained, we give a proof in Appendix B.

For compactness we have the following partially generalization of [36, Theorem
3.9]. The result follows by similar arguments as in the proof of [36, Theorem
3.9]. The details are left for the reader.

######  Proposition 1.20.

Let $\omega_{1},\omega_{2}\in\mathscr{P}_{E}(\mathbf{R}^{2d})$, and let
$\mathscr{B}$ be an invariant QBF-space on $\mathbf{R}^{2d}$ with respect to
$v(x,\xi)=1$ everywhere. Then the following conditions are equivalent:

  * •

at least one of the mappings in (1.25) is compact;

  * •

all the mappings in (1.25) are compact;

  * •

$\displaystyle{\lim_{|(x,\xi)|\to\infty}\frac{\omega_{2}(x,\xi)}{\omega_{1}(x,\xi)}=0}$.

######  Remark 1.21.

If $\omega\in\mathscr{P}_{E}(\mathbf{R}^{2d})$, then

$\displaystyle M_{0}(\omega,L^{p,q}(\mathbf{R}^{2d}))$ $\displaystyle\subseteq
M(\omega,L^{p,q}(\mathbf{R}^{2d}))=M^{p,q}_{(\omega)}(\mathbf{R}^{d}),$
$\displaystyle\quad p,q$ $\displaystyle\in(0,\infty],$ (1.26) and
$\displaystyle M_{0}(\omega,L^{p,q}(\mathbf{R}^{2d}))$
$\displaystyle=M(\omega,L^{p,q}(\mathbf{R}^{2d}))=M^{p,q}_{(\omega)}(\mathbf{R}^{d}),$
$\displaystyle\quad p,q$ $\displaystyle\in(0,\infty).$ (1.27)

In fact, (1.26) and the last two equalities in (1.27) follow from the fact
that $M^{p,q}_{(\omega)}(\mathbf{R}^{d})$ is continuously embedded in
$M^{\infty}_{(\omega)}(\mathbf{R}^{d})$ when $p,q\in(0,\infty]$. The first
equality in (1.27) follows from the fact that $\Sigma_{1}(\mathbf{R}^{d})$ is
dense in $M^{p,q}_{(\omega)}(\mathbf{R}^{d})$ when $p,q<\infty$. We also
observe that the inclusion in (1.26) is strict, if and only if $p=\infty$ or
$q=\infty$. (See e. g. [26, 28, 48].)

A broader family of modulation spaces is obtained by choosing $\mathscr{B}$ in
Definition 1.15 as the mixed quasi-Banach Orlicz space
$L^{\Phi,\Psi}(\mathbf{R}^{2d})$, where $\Phi$ and $\Psi$ are quasi-Young
functions (cf. e. g. [45, 53, 54]). In this case we obtain

$M_{0}(\omega,L^{\Phi,\Psi}(\mathbf{R}^{2d}))\subseteq
M(\omega,L^{\Phi,\Psi}(\mathbf{R}^{2d}))=M^{\Phi,\Psi}_{(\omega)}(\mathbf{R}^{d}),$
(1.28)

where $M^{\Phi,\Psi}_{(\omega)}(\mathbf{R}^{d})$ is the (quasi-Banach) Orlicz
modulation space with respect to $\Phi$, $\Psi$ and $\omega$. If
$p\in\mathbf{R}_{+}$,

$\Phi_{p}(t)=t^{p}\quad\text{and}\quad\Phi_{\infty}(t)=\begin{cases}0,&t\leq
1,\\\\[4.30554pt] \infty,&t>1,\end{cases}$

then $L^{\Phi_{p}}=L^{p}$, when $p\in(0,\infty]$. This in turn gives
$M^{\Phi_{p},\Phi_{q}}_{(\omega)}=M^{p,q}_{(\omega)}$. We observe that the
modulation spaces in (1.28) are normal, in view of [54]. We refer to [45, 53,
54] for more facts on Orlicz modulation spaces.

######  Remark 1.22.

Let $\omega\in\mathscr{P}_{E}(\mathbf{R}^{2d})$ and $\mathscr{B}$ be an
invariant QBF space. We observe that the only possible situation in which
$M(\omega,\mathscr{B})$ fails to be normal is when $\mathscr{B}$ is not a
Banach space because if $\mathscr{B}$ in Definition 1.15 is an invariant BF
space on $\mathbf{R}^{2d}$, then $M(\omega,\mathscr{B})$ is normal. (See e. g.
[36].)

##  2\. Convolution and twisted convolution factorizations of Wiener amalgam
spaces

In this section we apply Theorem 1.5 from the previous section to show that
Wiener amalgam spaces of the form $W\\!\\!L^{1,r}_{(v)}(\mathbf{R}^{d})$ for
$r\in(0,1]$ is a factorization algebra under convolution and twisted
convolution. Another application of Theorem 1.5 then leads to that
$W\\!\\!L^{p,q}_{(\omega)}(\mathbf{R}^{d})$ is a factorization module over
these algebras, when $p\geq 1$, $q\geq r$ and $\omega$ being $v$-moderate.

###  2.1. Convolution factorizations of Wiener amalgam spaces

Our main application of Theorem 1.5 for convolutions on Wiener amalgam spaces
is Theorem 2.2 below. In the following preparing result we ensure needed
continuity properties for Wiener amalgam spaces under convolution. Here the
involved weight functions should satisfy (1.13), and the constant

$\displaystyle c_{v}$ $\displaystyle=\sup_{x\in Q}(v(x),1),\ \quad
Q=[0,1]^{d}$ (2.1)

appears naturally.

######  Proposition 2.1.

Let $r\in(0,1]$, $p\in[1,\infty]$, $q\in[r,\infty]$ and
$\omega,v\in\mathscr{P}_{E}(\mathbf{R}^{d})$ be such that $\omega$ is
$v$-moderate. Then the map $(f,g)\mapsto f*g$ from
$\Sigma_{1}(\mathbf{R}^{d})\times\Sigma_{1}(\mathbf{R}^{d})$ to
$\Sigma_{1}(\mathbf{R}^{d})$ is uniquely extendable to a continuous and
commutative map from

$W\\!\\!L^{1,r}_{(v)}(\mathbf{R}^{d})\times
W\\!\\!L^{p,q}_{(\omega)}(\mathbf{R}^{d})\quad\text{or}\quad
W\\!\\!L^{p,q}_{(\omega)}(\mathbf{R}^{d})\times
W\\!\\!L^{1,r}_{(v)}(\mathbf{R}^{d})$

to $W\\!\\!L^{p,q}_{(\omega)}(\mathbf{R}^{d})$. If $c_{v}$ is given by (2.1),
then

$\displaystyle\|f*g\|_{W\\!\\!L^{p,q}_{(\omega)}}$
$\displaystyle=\|g*f\|_{W\\!\\!L^{p,q}_{(\omega)}}\leq
2^{d}c_{v}\|f\|_{W\\!\\!L^{1,r}_{(v)}}\|g\|_{W\\!\\!L^{p,q}_{(\omega)}},$
(2.2) $\displaystyle f$ $\displaystyle\in
W\\!\\!L^{1,r}_{(v)}(\mathbf{R}^{d}),\ g\in
W\\!\\!L^{p,q}_{(\omega)}(\mathbf{R}^{d}).$

Proposition 2.1 essentially shows that

$W\\!\\!L^{1,r}_{(v)}(\mathbf{R}^{d})*W\\!\\!L^{p,q}_{(\omega)}(\mathbf{R}^{d})=W\\!\\!L^{p,q}_{(\omega)}(\mathbf{R}^{d})*W\\!\\!L^{1,r}_{(v)}(\mathbf{R}^{d})\subseteq
W\\!\\!L^{p,q}_{(\omega)}(\mathbf{R}^{d}).$ (2.3)

A proof of Proposition 2.1 can be found in e. g. [26, Lemma 2.9] or [50,
Proposition 2.5]. (See also [28, 37] for related results.) In order to be self
contained we present a proof in Appendix C.

If in addition $p,q<\infty$, then the following result shows that (2.3) holds
with _equality_.

######  Theorem 2.2.

Let $r\in(0,1]$, $p\in[1,\infty)$, $q\in[r,\infty)$ and
$\omega,v\in\mathscr{P}_{E}(\mathbf{R}^{d})$ be such that $\omega$ is
$v$-moderate. Then

$W\\!\\!L^{1,r}_{(v)}(\mathbf{R}^{d})*W\\!\\!L^{p,q}_{(\omega)}(\mathbf{R}^{d})=W\\!\\!L^{p,q}_{(\omega)}(\mathbf{R}^{d})*W\\!\\!L^{1,r}_{(v)}(\mathbf{R}^{d})=W\\!\\!L^{p,q}_{(\omega)}(\mathbf{R}^{d}).$
(2.4)

We notice that Theorem 2.2 means that the mappings in Proposition 2.1 are
surjective when $p,q<\infty$.

###### Proof.

Since the convolution is commutative it suffices to prove that

$W\\!\\!L^{p,q}_{(\omega)}(\mathbf{R}^{d})*W\\!\\!L^{1,r}_{(v)}(\mathbf{R}^{d})=W\\!\\!L^{p,q}_{(\omega)}(\mathbf{R}^{d}).$

The assertion will follow from Theorem 1.5 and Proposition 2.1 if we show that
$\mathcal{B}$ possess a bounded (left) approximate identity for
$(\mathcal{B},\mathcal{M})$, with

$\mathcal{B}=W\\!\\!L^{1,r}_{(v)}(\mathbf{R}^{d})\quad\text{and}\quad\mathcal{M}=W\\!\\!L^{p,q}_{(\omega)}(\mathbf{R}^{d}).$

Hence, if $\phi\in C_{0}^{\infty}(\mathbf{R}^{d};[0,1])$ satisfies
$\int_{\mathbf{R}^{d}}\phi(y)\,dy=1$, it suffices to prove

$\|f-f*\phi_{\varepsilon}\|_{W\\!\\!L^{p,q}_{(\omega)}}\to
0\quad\text{as}\quad\varepsilon\to
0+,\quad\phi_{\varepsilon}(x)=\varepsilon^{-d}\phi(\varepsilon^{-1}x),\
\varepsilon>0,$ (2.5)

which is proven in similar ways as when dealing with Lebesgue spaces.

In fact, since $p,q<\infty$, it follows that $C_{0}^{\infty}(\mathbf{R}^{d})$
is dense in $W\\!\\!L^{p,q}_{(\omega)}(\mathbf{R}^{d})$, which implies that it
suffices to prove (2.5) when $f\in C_{0}^{\infty}(\mathbf{R}^{d})$.

Therefore, suppose that $f\in C_{0}^{\infty}(\mathbf{R}^{d})$, and let
$Q=[0,1]^{d}$. Then Minkowski’s inequality and the mean-value theorem give
that for some $y_{j}=y_{j}(\varepsilon)\in Q$, one has

$\displaystyle\|f-f$
$\displaystyle*\phi_{\varepsilon}\|_{W\\!\\!L^{p,q}_{(\omega)}}$
$\displaystyle=\left(\sum_{j\in\mathbf{Z}^{d}}\left(\int_{j+Q}\left|\int_{\mathbf{R}^{d}}(f(x)-f(x-\varepsilon
y))\phi(y)\,dy\right|^{p}\,dx\right)^{\frac{q}{p}}\omega(j)^{q}\right)^{\frac{1}{q}}$
$\displaystyle\leq\left(\sum_{j\in\mathbf{Z}^{d}}\left(\int_{\mathbf{R}^{d}}\left(\int_{j+Q}\left|(f(x)-f(x-\varepsilon
y))\omega(j)\right|^{p}\,dx\right)^{\frac{1}{p}}\phi(y)\,dy\right)^{q}\right)^{\frac{1}{q}}$
$\displaystyle\asymp\left(\sum_{j\in\mathbf{Z}^{d}}\left(\int_{j+Q}\left|(f(x)-f(x-\varepsilon
y_{j}(\varepsilon)))\omega(j)\right|^{p}\,dx\right)^{\frac{q}{p}}\right)^{\frac{1}{q}}\left(\int_{\mathbf{R}^{d}}\phi(y)\,dy\right)$

which tends to $0$ as $\varepsilon\to 0+$. Here observe that at most finite
numbers of terms in the sums are non-zero. This gives the result. ∎

By letting $q=p<\infty$ in the previous result we get the following.

######  Corollary 2.3.

Let $r\in(0,1]$, $p\in[1,\infty)$ and
$\omega,v\in\mathscr{P}_{E}(\mathbf{R}^{d})$ be such that $\omega$ is
$v$-moderate holds. Then

$W\\!\\!L^{1,r}_{(v)}(\mathbf{R}^{d})*L^{p}_{(\omega)}(\mathbf{R}^{d})=L^{p}_{(\omega)}(\mathbf{R}^{d})*W\\!\\!L^{1,r}_{(v)}(\mathbf{R}^{d})=L^{p}_{(\omega)}(\mathbf{R}^{d}).$

###  2.2. Twisted convolution factorizations of Wiener amalgam spaces

A Fourier transform and non-commutative convolution multiplication, connected
to the Weyl product in the theory of pseudo-differential operators concern the
symplectic Fourier transform and the twisted convolution. The _symplectic
Fourier transform_ of $\mathfrak{a}\in\Sigma_{1}(\mathbf{R}^{2d})$ is defined
by the formula

$(\mathscr{F}_{\sigma}\mathfrak{a})(X)=\pi^{-d}\int_{\mathbf{R}^{2d}}\mathfrak{a}(Y)e^{2i\sigma(X,Y)}\,dY.$
(2.6)

Here $\sigma$ from $\mathbf{R}^{2d}\times\mathbf{R}^{2d}$ to $\mathbf{R}$ is
the symplectic form, given by

$\sigma(X,Y)=\langle y,\xi\rangle-\langle x,\eta\rangle,\qquad
X=(x,\xi)\in\mathbf{R}^{2d},\ Y=(y,\eta)\in\mathbf{R}^{2d}.$

We note that

$\mathscr{F}_{\sigma}=T\circ(\mathscr{F}\otimes(\mathscr{F}^{-1})),\quad\text{when}\quad(T\mathfrak{a})(x,\xi)=2^{d}a(2\xi,2x).$

In particular, ${\mathscr{F}_{\sigma}}$ is continuous on
$\Sigma_{1}(\mathbf{R}^{2d})$, and extends uniquely to a homeomorphism on
$\Sigma_{1}^{\prime}(\mathbf{R}^{2d})$, and to a unitary map on
$L^{2}(\mathbf{R}^{2d})$, since similar facts hold for $\mathscr{F}$.
Furthermore, $\mathscr{F}_{\sigma}^{2}$ is the identity operator.

Let $\mathfrak{a},\mathfrak{b}\in\Sigma_{1}(\mathbf{R}^{2d})$. Then the
_twisted convolution_ of $\mathfrak{a}$ and $\mathfrak{b}$ is defined by the
formula

$(\mathfrak{a}\ast_{\sigma}\mathfrak{b})(X)=\left({{\frac{2}{\pi}}}\right)^{\frac{d}{2}}\int_{\mathbf{R}^{2d}}\mathfrak{a}(X-Y)\mathfrak{b}(Y)e^{2i\sigma(X,Y)}\,dY.$
(2.7)

The definition of $*_{\sigma}$ extends in different ways. For example, it
extends to a continuous multiplication on $L^{p}(\mathbf{R}^{2d})$ when
$p\in[1,2]$, and to a continuous map from
$\Sigma_{1}^{\prime}(\mathbf{R}^{2d})\times\Sigma_{1}(\mathbf{R}^{2d})$ to
$\Sigma_{1}^{\prime}(\mathbf{R}^{2d})$. We also remark that for the twisted
convolution we have

$\mathscr{F}_{\sigma}(\mathfrak{a}*_{\sigma}\mathfrak{b})=(\mathscr{F}_{\sigma}\mathfrak{a})*_{\sigma}\mathfrak{a}=\check{\mathfrak{a}}*_{\sigma}(\mathscr{F}_{\sigma}\mathfrak{b}),$
(2.8)

where $\check{\mathfrak{a}}(X)=\mathfrak{a}(-X)$ (cf. [23, 47, 49, 51]).

By

$|\mathfrak{a}*_{\sigma}\mathfrak{b}|\leq{\textstyle{\left(\frac{\pi}{2}\right)^{\frac{d}{2}}}}(|\mathfrak{a}|*|\mathfrak{b}|),$

similar arguments as in the proof of Proposition 2.1 give the following. The
details are left for the reader. Here the involved weight functions should
satisfy similar properties as before. More precisely, they should satisfy

$\displaystyle\omega(X+Y)$ $\displaystyle\leq\omega(X)v(Y),\qquad
X,Y\in\mathbf{R}^{2d},$ (2.9) and the constant $\displaystyle c_{v}$
$\displaystyle=\sup_{X\in Q}(v(X),1),\ \quad Q=[0,1]^{2d}$ (2.10)

is still important.

######  Proposition 2.4.

Let $r\in(0,1]$, $p\in[1,\infty]$, $q\in[r,\infty]$ and
$\omega,v\in\mathscr{P}_{E}(\mathbf{R}^{2d})$ be such that (2.9) holds. Then
the map $(\mathfrak{a},\mathfrak{b})\mapsto\mathfrak{a}*_{\sigma}\mathfrak{b}$
from $\Sigma_{1}(\mathbf{R}^{2d})\times\Sigma_{1}(\mathbf{R}^{2d})$ to
$\Sigma_{1}(\mathbf{R}^{2d})$ is uniquely extendable to continuous mappings
from

$W\\!\\!L^{1,r}_{(v)}(\mathbf{R}^{2d})\times
W\\!\\!L^{p,q}_{(\omega)}(\mathbf{R}^{2d})\quad\text{or}\quad
W\\!\\!L^{p,q}_{(\omega)}(\mathbf{R}^{2d})\times
W\\!\\!L^{1,r}_{(v)}(\mathbf{R}^{2d})$

to $W\\!\\!L^{p,q}_{(\omega)}(\mathbf{R}^{2d})$. If $c_{v}$ is given by
(2.10), then

$\displaystyle\max\big{(}\|\mathfrak{a}*_{\sigma}\mathfrak{b}\|_{W\\!\\!L^{p,q}_{(\omega)}},\|\mathfrak{b}*_{\sigma}\mathfrak{a}\|_{W\\!\\!L^{p,q}_{(\omega)}}\big{)}$
$\displaystyle\leq
4^{d}{\textstyle{\left(\frac{\pi}{2}\right)^{\frac{d}{2}}}}c_{v}\|\mathfrak{a}\|_{W\\!\\!L^{1,r}_{(v)}}\|\mathfrak{b}\|_{W\\!\\!L^{p,q}_{(\omega)}},$
(2.11) $\displaystyle\mathfrak{a}$ $\displaystyle\in
W\\!\\!L^{1,r}_{(v)}(\mathbf{R}^{2d}),\ \mathfrak{b}\in
W\\!\\!L^{p,q}_{(\omega)}(\mathbf{R}^{2d}).$

The following result shows that if $p,q<\infty$, then the mappings in
Proposition 2.4 are _surjective._ The result follows by similar arguments as
in the proof of Theorem 2.2, using Proposition 2.4 instead of Proposition 2.1.
The details are left for the reader.

######  Theorem 2.5.

Let $r\in(0,1]$, $p\in[1,\infty)$, $q\in[r,\infty)$ and
$\omega,v\in\mathscr{P}_{E}(\mathbf{R}^{2d})$ be such that $\omega$ is
$v$-moderate. Then

$W\\!\\!L^{1,r}_{(v)}(\mathbf{R}^{2d})*_{\sigma}W\\!\\!L^{p,q}_{(\omega)}(\mathbf{R}^{2d})=W\\!\\!L^{p,q}_{(\omega)}(\mathbf{R}^{2d})*_{\sigma}W\\!\\!L^{1,r}_{(v)}(\mathbf{R}^{2d})=W\\!\\!L^{p,q}_{(\omega)}(\mathbf{R}^{2d}).$
(2.12)

By letting $q=p$ in the previous result we get the following.

######  Corollary 2.6.

Let $r\in(0,1]$, $p\in[1,\infty)$ and
$\omega,v\in\mathscr{P}_{E}(\mathbf{R}^{2d})$ be such that (2.9) holds. Then

$W\\!\\!L^{1,r}_{(v)}(\mathbf{R}^{2d})*_{\sigma}L^{p}_{(\omega)}(\mathbf{R}^{2d})=L^{p}_{(\omega)}(\mathbf{R}^{2d})*_{\sigma}W\\!\\!L^{1,r}_{(v)}(\mathbf{R}^{2d})=L^{p}_{(\omega)}(\mathbf{R}^{2d}).$

######  Remark 2.7.

Suppose that $p\in[1,\infty)$. Since

$W\\!\\!L^{1,r}_{(v)}(\mathbf{R}^{d})\subsetneq L^{1}_{(v)}(\mathbf{R}^{d})$

when $r\in(0,1)$ and $\omega,v\in\mathscr{P}_{E}(\mathbf{R}^{d})$ are such
that $\omega$ is $v$-moderate, it follows that Corollary 2.3 is a strict
improvement of the weighted Rudin’s identity

$L^{1}_{(v)}(\mathbf{R}^{d})*L^{p}_{(\omega)}(\mathbf{R}^{d})=L^{p}_{(\omega)}(\mathbf{R}^{d})*L^{1}_{(v)}(\mathbf{R}^{d})=L^{p}_{(\omega)}(\mathbf{R}^{d}),$

which follows by choosing $r=1$ in that corollary. In the same way it follows
that Corollary 2.6 is a strict improvement of

$L^{1}_{(v)}(\mathbf{R}^{2d})*_{\sigma}L^{p}_{(\omega)}(\mathbf{R}^{2d})=L^{p}_{(\omega)}(\mathbf{R}^{2d})*_{\sigma}L^{1}_{(v)}(\mathbf{R}^{2d})=L^{p}_{(\omega)}(\mathbf{R}^{2d})$

when $\omega,v\in\mathscr{P}_{E}(\mathbf{R}^{2d})$ satisfy that $\omega$ is
$v$-moderate. (See also (0.9) in the introduction.)

######  Remark 2.8.

Suppose that $p=\infty$. Then we observe that the assertion in Theorem 2.2
fails to hold. In fact, using that $\Sigma_{1}(\mathbf{R}^{d})$ is dense in
$W\\!\\!L^{1,r}_{(v)}(\mathbf{R}^{d})$ it follows that

$W\\!\\!L^{1,r}_{(v)}(\mathbf{R}^{d})*W\\!\\!L^{\infty,q}_{(\omega)}(\mathbf{R}^{d})=W\\!\\!L^{\infty,q}_{(\omega)}(\mathbf{R}^{d})*W\\!\\!L^{1,r}_{(v)}(\mathbf{R}^{2d})\subseteq
C(\mathbf{R}^{d}),$

while

$W\\!\\!L^{\infty,q}_{(\omega)}(\mathbf{R}^{d})\nsubseteq C(\mathbf{R}^{d}).$

In similar ways it follows that the assertion in Theorem 2.5 fails to hold
when $p=\infty$.

For the moment we are not able to draw any conclusion wether the assertions in
Theorems 2.2 and 2.5 fail to hold when $p<\infty$ and $q=\infty$.

##  3\. Convolution factorizations of modulation spaces

In this section we apply Theorem 1.5 from Section A to show that with natural
assumptions on $\omega$, $v$ and $\mathscr{B}$, the modulation space
$M(\omega,\mathscr{B})$ is a factorization module over the Wiener amalgam
space $W\\!\\!L^{1,r}_{(v)}(\mathbf{R}^{d})$ for $r\in(0,1]$ under
convolution. In particular it follows that if $\omega$ is $v$-moderate and
$r\leq p,q<\infty$, then $M^{p,q}_{(\omega)}(\mathbf{R}^{d})$ and
$W^{p,q}_{(\omega)}(\mathbf{R}^{d})$ are factorization modules over
$W\\!\\!L^{1,r}_{(v)}(\mathbf{R}^{d})$.

In order to be more specific, we have the following.

######  Proposition 3.1.

Suppose $\omega,v_{1}\in\mathscr{P}_{E}(\mathbf{R}^{2d})$ are such that
$v_{1}$ is submultiplicative and $\omega$ is $v_{1}$-moderate, $\mathscr{B}$
is an invariant QBF space on $\mathbf{R}^{2d}$ with respect to
$v_{0}\in\mathscr{P}_{E}(\mathbf{R}^{2d})$ of order $r\in(0,1]$, and that
$M(\omega,\mathscr{B})$ is normal. Also let

$v(x)=v_{1}(x,0)v_{0}(x,0).$

Then the map $(f,g)\mapsto f*g$ from
$\Sigma_{1}(\mathbf{R}^{d})\times\Sigma_{1}(\mathbf{R}^{d})$ to
$\Sigma_{1}(\mathbf{R}^{d})$ is uniquely extendable to a continuous map from

$W\\!\\!L^{1,r}_{(v)}(\mathbf{R}^{d})\times
M(\omega,\mathscr{B})\quad\text{or}\quad M(\omega,\mathscr{B})\times
W\\!\\!L^{1,r}_{(v)}(\mathbf{R}^{d})$

to $M(\omega,\mathscr{B})$. Furthermore,

$\displaystyle\|f*g\|_{M(\omega,\mathscr{B})}$
$\displaystyle=\|g*f\|_{M(\omega,\mathscr{B})}\lesssim\|f\|_{W\\!\\!L^{1,r}_{(v)}}\|g\|_{M(\omega,\mathscr{B})},$
(3.1) $\displaystyle f$ $\displaystyle\in
W\\!\\!L^{1,r}_{(v)}(\mathbf{R}^{d}),\quad g\in M(\omega,\mathscr{B}).$

We observe that Proposition 3.1 essentially means that

$W\\!\\!L^{1,r}_{(v)}(\mathbf{R}^{d})*M(\omega,\mathscr{B})=M(\omega,\mathscr{B})*W\\!\\!L^{1,r}_{(v)}(\mathbf{R}^{d})\subseteq
M(\omega,\mathscr{B}).$ (3.2)

###### Proof.

We replace the weights $\omega$, $v$, $v_{0}$ and $v_{1}$ with equivalent
smooth weights, which is possible in view of e. g. [49]. First suppose that
$f\in\Sigma_{1}(\mathbf{R}^{d})$ and $g\in M(\omega,\mathscr{B})$. Let
$Q=[0,1]^{d}$

$G(x,\xi)=|\mathcal{T}_{\phi}g(x,\xi)\omega(x,\xi)|\quad\text{and}\quad
H(x,\xi)=\int_{\mathbf{R}^{d}}|f(y)v_{1}(y,0)|G(x-y,\xi)\,dy.$

Then $G$ and $H$ are continuous, and (1.16) and the fact that $\omega$ is
$v_{1}$ moderate give

$|(\mathcal{T}_{\phi}(f*g))(x,\xi)\omega(x,\xi)|\lesssim H(x,\xi).$

By the mean-value theorem, Minkowski’s inequality, (1.20) and (1.16) give for
some $y_{j}\in Q$, $j\in\mathbf{Z}^{d}$, that

$\displaystyle\|f*g\|_{M(\omega,\mathscr{B})}^{r}$
$\displaystyle\lesssim\|H\|_{\mathscr{B}}^{r}=\left\|\sum_{j\in\mathbf{Z}^{d}}\int_{j+Q}|f(y)v_{1}(y,0)|G\big{(}\,\cdot\,-(y,0)\big{)}\,dy\right\|_{\mathscr{B}}^{r}$
$\displaystyle=\left\|\sum_{j\in\mathbf{Z}^{d}}G\big{(}\,\cdot\,-(j+y_{j},0)\big{)}\int_{j+Q}|f(y)v_{1}(y,0)|\,dy\right\|_{\mathscr{B}}^{r}$
$\displaystyle\leq\sum_{j\in\mathbf{Z}^{d}}\left\|G\big{(}\,\cdot\,-(j+y_{j},0)\big{)}\right\|_{\mathscr{B}}^{r}\left(\int_{j+Q}|f(y)v_{1}(y,0)|\,dy\right)^{r}$
$\displaystyle\lesssim\sum_{j\in\mathbf{Z}^{d}}\left(\int_{j+Q}|f(y)v_{1}(y,0)v_{0}(j+y_{j},0)|\,dy\right)^{r}\|G\|_{\mathscr{B}}^{r}$
$\displaystyle\asymp\sum_{j\in\mathbf{Z}^{d}}\left(\int_{j+Q}|f(y)v_{1}(y,0)v_{0}(y,0)|\,dy\right)^{r}\|G\|_{\mathscr{B}}^{r}$
$\displaystyle=\|f\|_{W\\!\\!L^{1,r}_{(v)}}^{r}\|G\|_{\mathscr{B}}^{r},$

which shows that

$\|f*g\|_{M(\omega,\mathscr{B})}\lesssim\|f\|_{W\\!\\!L^{1,r}_{(v)}},\|g\|_{M(\omega,\mathscr{B})}$

and (3.1) follows in the case when $f\in\Sigma_{1}(\mathbf{R}^{d})$. The
assertion now follows in general from the facts that (3.1) holds with
$f\in\Sigma_{1}(\mathbf{R}^{d})$ and that $\Sigma_{1}(\mathbf{R}^{d})$ is
dense in $W\\!\\!L^{1,r}_{(v)}(\mathbf{R}^{d})$. ∎

######  Remark 3.2.

Evidently, Proposition 3.1 holds true with $M_{0}(\omega,\mathscr{B})$ in
place of $M_{0}(\omega,\mathscr{B})$ at each occurrence. In particular,

$W\\!\\!L^{1,r}_{(v)}(\mathbf{R}^{d})*M_{0}(\omega,\mathscr{B})=M_{0}(\omega,\mathscr{B})*W\\!\\!L^{1,r}_{(v)}(\mathbf{R}^{d})\subseteq
M_{0}(\omega,\mathscr{B}).$ (3.2)′

The following result shows that (3.2)′ holds with equality.

######  Theorem 3.3.

$\omega,v_{1}\in\mathscr{P}_{E}(\mathbf{R}^{2d})$ are such that $v_{1}$ is
submultiplicative and $\omega$ is $v_{1}$-moderate, and suppose that
$\mathscr{B}$ is an invariant QBF space on $\mathbf{R}^{2d}$ with respect to
$v_{0}\in\mathscr{P}_{E}(\mathbf{R}^{2d})$ of order $r\in(0,1]$. Also let

$v(x)=v_{1}(x,0)v_{0}(x,0).$

Then

$W\\!\\!L^{1,r}_{(v)}(\mathbf{R}^{d})*M_{0}(\omega,\mathscr{B})=M_{0}(\omega,\mathscr{B})*W\\!\\!L^{1,r}_{(v)}(\mathbf{R}^{d})=M_{0}(\omega,\mathscr{B}).$
(3.2)′′

###### Proof.

Since $\Sigma_{1}(\mathbf{R}^{d})$ is dense in $M_{0}(\omega,\mathscr{B})$, it
follows from Theorem 1.5 that the result follows if we prove that

$\lim_{\varepsilon\to
0}\|f*\psi_{\varepsilon}-f\|_{M(\omega,\mathscr{B})}=0,\qquad
f\in\Sigma_{1}(\mathbf{R}^{d}),$ (3.3)

when $\psi_{\varepsilon}=\varepsilon^{-d}\psi(\varepsilon^{-1}\,\cdot\,)$,
where $\psi\in\Sigma_{1}(\mathbf{R}^{d})$ satisfies

$\int_{\mathbf{R}^{d}}\psi(x)\,dx=1.$

For $f\in\Sigma_{1}(\mathbf{R}^{d})$ and $v_{2}=v\cdot v_{1}$ we have

$\displaystyle\|f(\,\cdot\,-x)e^{i\langle\,\cdot\,,\xi\rangle}\|_{M(\omega,\mathscr{B})}$
$\displaystyle\asymp\|V_{\phi}f(\,\cdot\,-(x,\xi))\cdot\omega\|_{\mathscr{B}}$
$\displaystyle\lesssim\|V_{\phi}f(\,\cdot\,-(x,\xi))\cdot\omega(\,\cdot\,-(x,\xi))\|_{\mathscr{B}}v_{1}(x,\xi)$
$\displaystyle\lesssim\|V_{\phi}f\cdot\omega\|_{\mathscr{B}}v(x,\xi)v_{1}(x,\xi)=\|f\|_{M(\omega,\mathscr{B})}v_{2}(x,\xi).$

By Feichtinger’s minimization principle for translation and modulation
invariant quasi-Banach spaces it follows that
$M^{r,r}_{(v_{2})}(\mathbf{R}^{d})\subseteq M(\omega,\mathscr{B})$ and that

$\|f\|_{M(\omega,\mathscr{B})}\lesssim\|f\|_{M^{r,r}_{(v_{2})}},\qquad f\in
M^{r,r}_{(v_{2})}(\mathbf{R}^{d}).$

(See [52, Theorem 2.4].) This gives

$\displaystyle\|f*\psi_{\varepsilon}-f\|_{M(\omega,\mathscr{B})}^{r}\lesssim\|f*\psi_{\varepsilon}-f\|_{M^{r,r}_{(v_{2})}}^{r}$
$\displaystyle\lesssim\iint_{\mathbf{R}^{2d}}\left(\int_{\mathbf{R}^{d}}|V_{\phi}f(x-\varepsilon
y,\xi)-V_{\phi}f(x,\xi)|\,|\psi(y)|v_{2}(x,\xi)\,dy\right)^{r}\,dxd\xi$

Since $v_{2}\in\mathscr{P}_{E}(\mathbf{R}^{2d})$,
$V_{\phi}f\in\Sigma_{1}(\mathbf{R}^{2d})$ and $\psi\in\Sigma(\mathbf{R}^{d})$,
it follows that for some $r_{0}\geq 0$ we have

$\displaystyle v_{2}(x,\xi)$ $\displaystyle\lesssim e^{r_{0}(|x|+|\xi|)},$
$\displaystyle|V_{\phi}f(x,\xi)|$ $\displaystyle\lesssim e^{-r(|x|+|\xi|)}$
and $\displaystyle|\psi(x)|$ $\displaystyle\lesssim e^{-r|x|},$

for every $r>0$. This implies that

$H_{\varepsilon}(x,y,\xi)\equiv|V_{\phi}f(x-\varepsilon
y,\xi)-V_{\phi}f(x,\xi)|\,|\psi(y)|v_{2}(x,\xi)\\\\[4.30554pt] \leq
Ce^{-r(|x|+|y|+|\xi|)}\in L^{1}(\mathbf{R}^{3d}),$

where the constant $C>0$ is independent of $\varepsilon\in(0,1]$. Since
$H_{\varepsilon}(x,y,\xi)$ tends to $0$ pointwise as $\varepsilon\to 0+$,
Lebesgue’s theorem gives

$0\leq\|f*\psi_{\varepsilon}-f\|_{M(\omega,\mathscr{B})}^{r}\lesssim\iint_{\mathbf{R}^{2d}}\left(\int_{\mathbf{R}^{d}}H_{\varepsilon}(x,y,\xi)\,dy\right)^{r}\,dxd\xi\to
0$

as $\varepsilon\to 0+$. This gives (3.3) and the result follows. ∎

######  Corollary 3.4.

Suppose $p,q,r\in(0,\infty)$ satisfy $r\leq\min(p,q,1)$, and let
$\omega,v\in\mathscr{P}_{E}(\mathbf{R}^{2d})$ be such that $\omega$ is
$v$-moderate. Then

$W\\!\\!L^{1,r}_{(v)}(\mathbf{R}^{d})*M^{p,q}_{(\omega)}(\mathbf{R}^{d})=M^{p,q}_{(\omega)}(\mathbf{R}^{d})$

and

$W\\!\\!L^{1,r}_{(v)}(\mathbf{R}^{d})*W^{p,q}_{(\omega)}(\mathbf{R}^{d})=W^{p,q}_{(\omega)}(\mathbf{R}^{d}).$

By choosing $r=1$ in the previous corollary we obtain the following.

######  Corollary 3.5.

Suppose $p,q\in[1,\infty)$, and let
$\omega,v\in\mathscr{P}_{E}(\mathbf{R}^{2d})$ be such that $\omega$ is
$v$-moderate. Then

$L^{1}_{(v)}(\mathbf{R}^{d})*M^{p,q}_{(\omega)}(\mathbf{R}^{d})=M^{p,q}_{(\omega)}(\mathbf{R}^{d})$

and

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

##  4\. Convolution and twisted convolution factorizations of Schatten-von
Neumann symbols in pseudo-differential calculus

In this section we first recall in Subsections 4.1–4.3 some facts about
pseudo-differential operators, their Schatten-von Neumann symbol classes and
properties on compositions of such operators on symbol levels and their links
to twisted convolution. Thereafter we apply Theorem 1.5 from Section A to show
that such symbol classes are factorization modules over the Wiener amalgam
spaces $W\\!\\!L^{1,r}(\mathbf{R}^{2d})$. In the last part we apply Theorem
1.5 to show that these symbol classes are factorization modules over the space
of all symbols giving rise to compact operators.

###  4.1. Schatten-von Neumann classes

The definition of Schatten-von Neumann classes is often performed in
background of singular values (cf. e. g. [46, 8, 49]). Let $\mathcal{H}_{1}$
and $\mathcal{H}_{2}$ be Hilbert spaces and let $T$ be a linear operator from
$\mathcal{H}_{1}$ to $\mathcal{H}_{2}$. The singular value of $T$ of order
$j\geq 1$ is defined as

$\sigma_{j}(T)=\sigma_{j}(T;\mathcal{H}_{1},\mathcal{H}_{2})\equiv\inf\|T-T_{0}\|_{\mathcal{H}_{1}\to\mathcal{H}_{2}},$

where the infimum is taken over all linear operators $T_{0}$ from
$\mathcal{H}_{1}$ to $\mathcal{H}_{2}$ of rank at most $j-1$. The operator $T$
is said to be a Schatten-von Neumann operator of order $p\in(0,\infty]$ if

$\|T\|_{\mathscr{I}_{p}(\mathcal{H}_{1},\mathcal{H}_{2})}\equiv\|\\{\sigma_{j}(T)\\}_{j=1}^{\infty}\|_{\ell^{p}(\mathbf{Z}_{+})}$
(4.1)

is finite. The set of Schatten-von Neumann operators from $\mathcal{H}_{1}$ to
$\mathcal{H}_{2}$ of order $p\in(0,\infty]$ is denoted by
$\mathscr{I}_{p}(\mathcal{H}_{1},\mathcal{H}_{2})$.

We observe that if $p<\infty$, then
$\mathscr{I}_{p}(\mathcal{H}_{1},\mathcal{H}_{2})$ is contained in
$\mathcal{K}(\mathcal{H}_{1},\mathcal{H}_{2})$, the set of compact operators
from $\mathcal{H}_{1}$ to $\mathcal{H}_{2}$. Furthermore,
$\mathscr{I}_{\infty}(\mathcal{H}_{1},\mathcal{H}_{2})$ agrees with the set of
linear bounded operators from $\mathcal{H}_{1}$ to $\mathcal{H}_{2}$, with
equality in norms. We also have that
$\mathscr{I}_{1}(\mathcal{H}_{1},\mathcal{H}_{2})$ and
$\mathscr{I}_{2}(\mathcal{H}_{1},\mathcal{H}_{2})$ are the Banach spaces which
consist of all trace-class and Hilbert-Schmidt operators, respectively, from
$\mathcal{H}_{1}$ to $\mathcal{H}_{2}$, with equality in norms.

It is evident that $\mathscr{I}_{p}(\mathcal{H}_{1},\mathcal{H}_{2})$ is
contained in the set of linear and compact operators from $\mathcal{H}_{1}$ to
$\mathcal{H}_{2}$ when $p<\infty$. If we set
$\mathscr{I}_{\sharp}(\mathcal{H}_{1},\mathcal{H}_{2})\equiv\mathcal{K}(\mathcal{H}_{1},\mathcal{H}_{2})$,
and let $\mathbf{R}_{\sharp}=\mathbf{R}_{+}\cup\\{\sharp,\infty\\}$ with
orderings

$x<\sharp,\quad
x\leq\sharp,\quad\sharp<\infty\quad\text{and}\quad\sharp\leq\infty\quad\text{when}\quad
x\in\mathbf{R}_{+},$

then we get

$\begin{gathered}\mathscr{I}_{p_{1}}(\mathcal{H}_{1},\mathcal{H}_{2})\subseteq\mathscr{I}_{p_{2}}(\mathcal{H}_{1},\mathcal{H}_{2})\quad\text{and}\quad\|T\|_{\mathscr{I}_{p_{2}}(\mathcal{H}_{1},\mathcal{H}_{2})}\leq\|T\|_{\mathscr{I}_{p_{1}}(\mathcal{H}_{1},\mathcal{H}_{2})}\\\\[4.30554pt]
\text{when}\quad p_{1},p_{2}\in\mathbf{R}_{\sharp},\quad p_{1}\leq p_{2},\quad
T\in\mathscr{I}_{\infty}(\mathcal{H}_{1},\mathcal{H}_{2}).\end{gathered}$
(4.2)

Here recall that

$\|T\|_{\mathscr{I}_{\sharp}(\mathcal{H}_{1},\mathcal{H}_{2})}=\|T\|_{\mathcal{K}(\mathcal{H}_{1},\mathcal{H}_{2})}=\|T\|_{\mathcal{H}_{1}\mapsto\mathcal{H}_{2}}=\|T\|_{\mathscr{I}_{\infty}(\mathcal{H}_{1},\mathcal{H}_{2})},\quad
T\in\mathcal{K}(\mathcal{H}_{1},\mathcal{H}_{2})$ (4.3)

It follows that if $p\in\mathbf{R}_{\sharp}$, then
$\mathscr{I}_{p}(\mathcal{H}_{1},\mathcal{H}_{2})$ is a quasi-Banach space of
order $\min(1,p)$. In particular,
$\mathscr{I}_{p}(\mathcal{H}_{1},\mathcal{H}_{2})$ is a Banach space when
$p\geq 1$. (See e. g. [46].)

As a consequence of the spectral theorem we have the following (see e. g.
[46]). Here $\operatorname{ON}(\mathcal{H})$ is the set of all orthonormal
sequences in the Hilbert space $\mathcal{H}$. We also let
$\ell^{\sharp}(\mathbf{Z}_{+})$ be the set of all sequences in
$\ell^{\infty}(\mathbf{Z}_{+})$ which tend to zero at infinity.

######  Proposition 4.1.

Let $p\in\mathbf{R}_{\sharp}$ with $p\neq\infty$, $\mathcal{H}_{1}$,
$\mathcal{H}_{2}$ be Hilbert spaces and let
$T\in\mathscr{I}_{\sharp}(\mathcal{H}_{1},\mathcal{H}_{2})$. Then the
following is true:

  1. (1)

for some $\\{f_{j}\\}_{j=1}^{\infty}\in\operatorname{ON}(\mathcal{H}_{1})$ and
$\\{g_{j}\\}_{j=1}^{\infty}\in\operatorname{ON}(\mathcal{H}_{2})$ and unique
non-negative decreasing sequence $\\{\lambda_{j}\\}_{j=1}^{\infty}$, it holds

$Tf=\sum_{j=1}^{\infty}\lambda_{j}(f,f_{j})_{\mathcal{H}_{1}}g_{j},\qquad
f\in\mathcal{H}_{1},$ (4.4)

where the sum converges with respect to the $\mathcal{H}_{2}$ norm;

  2. (2)

if $\lambda_{j}$ are the same as in (4.4), then $\lambda_{j}=\sigma_{j}(T)$;

  3. (3)

$T\in\mathscr{I}_{p}(\mathcal{H}_{1},\mathcal{H}_{2})$, if and only if
$\\{\lambda_{j}\\}_{j=1}^{\infty}\in\ell^{p}(\mathbf{Z}_{+})$, and

$\|T\|_{\mathscr{I}_{p}(\mathcal{H}_{1},\mathcal{H}_{2})}=\|\\{\lambda_{j}\\}_{j=1}^{\infty}\|_{\ell^{p}(\mathbf{Z}_{+})}$

We also have the following Hölder’s inequality for compositions of Schatten-
von Neumann classes. Again we refer to [46] for the proof.

######  Proposition 4.2.

Let $p_{0},p_{1},p_{2}\in\mathbf{R}_{\sharp}$ be such that

$\frac{1}{p_{0}}\leq\frac{1}{p_{1}}+\frac{1}{p_{2}}\quad\text{and}\quad(p_{0},p_{1},p_{2})\neq(\sharp,\infty,\infty),$

and let $\mathcal{H}_{j}$, $j=0,1,2$, be Hilbert spaces. Then the map
$(T_{1},T_{2})\mapsto T_{2}\circ T_{1}$ restricts to a continuous map from

$\mathscr{I}_{p_{1}}(\mathcal{H}_{0},\mathcal{H}_{1})\times\mathscr{I}_{p_{2}}(\mathcal{H}_{1},\mathcal{H}_{2})\quad\text{to}\quad\mathscr{I}_{p_{0}}(\mathcal{H}_{0},\mathcal{H}_{2}),$

and

$\|T_{2}\circ
T_{1}\|_{\mathscr{I}_{p_{0}}(\mathcal{H}_{0},\mathcal{H}_{2})}\leq\|T_{1}\|_{\mathscr{I}_{p_{1}}(\mathcal{H}_{0},\mathcal{H}_{1})}\|T_{2}\|_{\mathscr{I}_{p_{2}}(\mathcal{H}_{1},\mathcal{H}_{2})},\\\\[4.30554pt]
T_{1}\in\mathscr{I}_{p_{1}}(\mathcal{H}_{0},\mathcal{H}_{1}),\
T_{2}\in\mathscr{I}_{p_{2}}(\mathcal{H}_{1},\mathcal{H}_{2}).$

###  4.2. Basic properties for pseudo-differential operators and Schatten-von
Neumann symbols

Recall that for any matrix $A\in\mathbf{M}(d,\mathbf{R})$, and
$\mathfrak{a}\in\Sigma_{1}^{\prime}(\mathbf{R}^{2d})$, the pseudo-differential
operator $\operatorname{Op}_{A}(\mathfrak{a})$ is the linear and continuous
mapping from $\Sigma_{1}(\mathbf{R}^{d})$ to
$\Sigma_{1}^{\prime}(\mathbf{R}^{d})$, given by the formula

$\operatorname{Op}_{A}(\mathfrak{a})f(x)\equiv(2\pi)^{-{2d}}\iint_{\mathbf{R}^{2d}}\mathfrak{a}(x-A(x-y),\xi)f(y)e^{i\langle
x-y,\xi\rangle}\,dyd\xi.$ (4.5)

Here the integrals shall be interpreted in distribution sense when
$\mathfrak{a}$ fails to belong to $L^{1}(\mathbf{R}^{2d})$. That is, for
general $\mathfrak{a}\in\Sigma_{1}^{\prime}(\mathbf{R}^{2d})$, (4.5) should be
interpreted as

$\displaystyle\operatorname{Op}_{A}(\mathfrak{a})f(x)$
$\displaystyle\equiv(2\pi)^{-\frac{d}{2}}\langle
K_{\mathfrak{a},A}(x,\,\cdot\,),f\rangle\quad\text{with}$ (4.5)′
$\displaystyle K_{\mathfrak{a},A}(x,y)$
$\displaystyle\equiv(\mathscr{F}_{2}^{-1}\mathfrak{a})(x-A(x-y),x-y),$

where $\mathscr{F}_{2}F$ is the partial Fourier transform of $F(x,y)$ with
respect to the $y$-variable. We observe that the map $\mathfrak{a}\mapsto
K_{\mathfrak{a},A}$ is homeomorphic on

$\Sigma_{1}(\mathbf{R}^{2d}),\quad\mathscr{S}(\mathbf{R}^{2d}),\quad\mathscr{S}^{\prime}(\mathbf{R}^{2d}),\quad\quad\text{and
on}\quad\Sigma_{1}^{\prime}(\mathbf{R}^{2d}),$

since similar facts hold true for $\mathscr{F}_{2}$ and the map

$F(x,y)\mapsto F(x-A(x-y),x-y).$

We observe that if $A=0$, then
$\operatorname{Op}_{A}(\mathfrak{a})=\operatorname{Op}_{0}(\mathfrak{a})$ is
the same as normal or Kohn-Nirenberg representation $\mathfrak{a}(x,D)$ of
$\mathfrak{a}\in\Sigma_{1}^{\prime}(\mathbf{R}^{2d})$. If instead
$A=\frac{1}{2}\cdot I_{d}$, where $I_{d}$ is the $d\times d$ unit matrix, then
$\operatorname{Op}_{A}(\mathfrak{a})$ is the Weyl quantization
$\operatorname{Op}^{w}(\mathfrak{a})$ of
$\mathfrak{a}\in\Sigma_{1}^{\prime}(\mathbf{R}^{2d})$.

A combination of the previous homeomorphisms and Schwartz kernel theorem shows
that the map

$\mathfrak{a}\mapsto\operatorname{Op}_{A}(\mathfrak{a})$ (4.6)

is bijective from $\Sigma_{1}^{\prime}(\mathbf{R}^{2d})$ and the set of linear
and continuous operators from $\Sigma_{1}(\mathbf{R}^{d})$ to
$\Sigma_{1}^{\prime}(\mathbf{R}^{d})$. The same holds true with $\mathscr{S}$
in place of $\Sigma_{1}$ at each occurrence. In particular it follows that if
$A,B\in\mathbf{M}(d,\mathbf{R})$ and
$\mathfrak{a}\in\Sigma_{1}^{\prime}(\mathbf{R}^{2d})$, then there is a unique
$\mathfrak{b}\in\Sigma_{1}^{\prime}(\mathbf{R}^{2d})$ such that
$\operatorname{Op}_{A}(\mathfrak{a})=\operatorname{Op}_{B}(\mathfrak{b})$. By
straight-forward applications of Fourier’s inversion formula it follows that

$\operatorname{Op}_{A}(\mathfrak{a})=\operatorname{Op}_{B}(\mathfrak{b})\qquad\Leftrightarrow\qquad
e^{i\langle AD_{\xi},D_{x}\rangle}\mathfrak{a}=e^{i\langle
BD_{\xi},D_{x}\rangle}\mathfrak{b}.$ (4.7)

(See also e. g. [51].)

If more restrictive $\mathfrak{a}\in\Sigma_{1}(\mathbf{R}^{2d})$, then
$\operatorname{Op}_{A}(\mathfrak{a})$ is continuous from
$\Sigma_{1}(\mathbf{R}^{d})$ to $\Sigma_{1}(\mathbf{R}^{d})$, and is uniquely
extendable to a continuous map from $\Sigma_{1}^{\prime}(\mathbf{R}^{d})$ to
$\Sigma_{1}(\mathbf{R}^{d})$.

For $p\in\mathbf{R}_{\sharp}$, we let $s_{p,A}(\mathbf{R}^{2d})$ be the set of
all $\mathfrak{a}\in\Sigma_{1}^{\prime}(\mathbf{R}^{2d})$ such that
$\operatorname{Op}_{A}(\mathfrak{a})$ belongs to
$\mathscr{I}_{p}=\mathscr{I}_{p}(L^{2}(\mathbf{R}^{d}))$. We equip
$s_{p,A}(\mathbf{R}^{2d})$ with the quasi-norm

$\|\mathfrak{a}\|_{s_{p,A}}\equiv\begin{cases}\|\operatorname{Op}_{A}(\mathfrak{a})\|_{\mathscr{I}_{p}},&p\neq\sharp\\\\[4.30554pt]
\|\operatorname{Op}_{A}(\mathfrak{a})\|_{\mathscr{I}_{\infty}},&p=\sharp,\end{cases}$

when $\mathfrak{a}\in s_{p,A}(\mathbf{R}^{2d})$.

It follows from the kernel theorem by Schwartz that the map (4.6) is an
isometric isomorphism from $s_{p,A}(\mathbf{R}^{2d})$ to $\mathscr{I}_{p}$. In
particular, $s_{p,A}(\mathbf{R}^{2d})$ is a quasi-Banach space under the
quasi-norm $\|\,\cdot\,\|_{s_{p,A}}$, which is a Banach space when $p\geq 1$.
Since $\mathscr{I}_{p}$ is strictly increasing with $p$ we get

$s_{p_{1},A}(\mathbf{R}^{2d})\subsetneq
s_{p_{2},A}(\mathbf{R}^{2d}),\quad\text{when}\quad
p_{1},p_{2}\in\mathbf{R}_{\sharp},\ p_{1}<p_{2}.$

We also have that

$\displaystyle\operatorname{Op}_{A}(s_{1,A}(\mathbf{R}^{2d})),$
$\displaystyle\operatorname{Op}_{A}(s_{2,A}(\mathbf{R}^{2d})),$
$\displaystyle\operatorname{Op}_{A}(s_{\sharp,A}(\mathbf{R}^{2d}))$ and
$\displaystyle\operatorname{Op}_{A}(s_{\infty,A}(\mathbf{R}^{2d}))$

are the sets of trace-class, Hilbert-Schmidt, compact and continuous
operators, respectively, on $L^{2}(\mathbf{R}^{d})$.

In the remaining part of the subsection we describe some Schatten-von Neumann
properties given in [46] in the framework of symbol calculus for pseudo-
differential operators. We mainly follow the approaches in [47, 51].

The spectral resolution of elements in $s_{p}^{A}(\mathbf{R}^{2d})$ can be
described in terms of $A$-Wigner distributions. For any
$A\in\mathbf{M}(d,\mathbf{R})$, the $A$-Wigner distribution of
$f,g\in\Sigma_{1}^{\prime}(\mathbf{R}^{d})$ is defined by

$W_{f,g}^{A}(x,\xi)\equiv\mathscr{F}\big{(}f(x+A\,\cdot\,)\overline{g(x+(A-I)\,\cdot\,)}\big{)}(\xi),$

which makes sense as an element in $\Sigma_{1}^{\prime}(\mathbf{R}^{2d})$. If
$f,g\in\Sigma_{1}(\mathbf{R}^{d})$, then $W_{f,g}^{A}(x,\xi)$ is given by

$W_{f,g}^{A}(x,\xi)=(2\pi)^{-\frac{d}{2}}\int_{\mathbf{R}^{d}}f(x+Ay)\overline{g(x+(A-I)y)}e^{-i\langle
y,\xi\rangle}\,dy,$

which becomes an element in $\Sigma_{1}(\mathbf{R}^{2d})$.

The following proposition describes fundamental links between Wigner
distributions and pseudo-differential operators. Here the symbol product
$\mathfrak{a}_{1}{\text{\footnotesize$\\#$}}_{A}\mathfrak{a}_{2}$ of
$\mathfrak{a}_{1},\mathfrak{a}_{2}\in\Sigma_{1}^{\prime}(\mathbf{R}^{2d})$ is
defined by the formula

$\operatorname{Op}_{A}(\mathfrak{a}_{1}{\text{\footnotesize$\\#$}}_{A}\mathfrak{a}_{2})=\operatorname{Op}_{A}(\mathfrak{a}_{1})\circ\operatorname{Op}_{A}(\mathfrak{a}_{2}),$
(4.8)

provided the right-hand side makes sense as a continuous operator from
$\Sigma_{1}(\mathbf{R}^{d})$ to $\Sigma_{1}^{\prime}(\mathbf{R}^{d})$. The
result follows by straight-forward computations. The details are left for the
reader. (See also [49, 51].)

######  Proposition 4.3.

Let $A\in\mathbf{M}(d,\mathbf{R})$ and
$\mathfrak{a}\in\Sigma_{1}^{\prime}(\mathbf{R}^{2d})$. Then the following is
true:

  1. (1)

if $f,g\in\Sigma_{1}(\mathbf{R}^{d})$ then

$(\operatorname{Op}_{A}(\mathfrak{a})f,g)_{L^{2}(\mathbf{R}^{2d})}=(2\pi)^{-\frac{d}{2}}(\mathfrak{a},W_{g,f}^{A})_{L^{2}(\mathbf{R}^{2d})}\text{;}$

  2. (2)

if $f\in\Sigma_{1}(\mathbf{R}^{d})$ and
$g\in\Sigma_{1}^{\prime}(\mathbf{R}^{d})$, then

$\mathfrak{a}{\text{\footnotesize$\\#$}}_{A}W_{f,g}^{A}=W_{\operatorname{Op}_{A}(\mathfrak{a})f,g}^{A}\text{;}$

  3. (3)

if $f\in\Sigma_{1}(\mathbf{R}^{d})$ and
$f_{0},g\in\Sigma_{1}^{\prime}(\mathbf{R}^{d})$, then

$\operatorname{Op}_{A}(W_{f_{0},g}^{A})f=(2\pi)^{-\frac{d}{2}}(f,g)_{L^{2}}f_{0}.$

A combination of (4.7) and Proposition 4.3 gives

$e^{i\langle AD_{\xi},D_{x}\rangle}W_{f,g}^{A}=e^{i\langle
BD_{\xi},D_{x}\rangle}W_{f,g}^{B},\qquad
f,g\in\Sigma_{1}^{\prime}(\mathbf{R}^{d}).$ (4.9)

We also observe that by straight-forward combination of (4.7) and (4.8) we get

$\big{(}e^{-i\langle
AD_{\xi},D_{x}\rangle}\mathfrak{c}_{1}\big{)}{\text{\footnotesize$\\#$}}_{A}\big{(}e^{-i\langle
AD_{\xi},D_{x}\rangle}\mathfrak{c}_{2}\big{)}\\\\[4.30554pt]
=\big{(}e^{-i\langle
BD_{\xi},D_{x}\rangle}\mathfrak{c}_{1}\big{)}{\text{\footnotesize$\\#$}}_{B}\big{(}e^{-i\langle
BD_{\xi},D_{x}\rangle}\mathfrak{c}_{2}\big{)},$ (4.10)

when
$\mathfrak{c}_{1},\mathfrak{c}_{2}\in\Sigma_{1}^{\prime}(\mathbf{R}^{2d})$,
provided the symbol products make sense as elements in
$\Sigma_{1}^{\prime}(\mathbf{R}^{2d})$.

Recall that $\operatorname{ON}=\operatorname{ON}(L^{2}(\mathbf{R}^{d}))$ is
the set of all orthonormal sequences $\\{f_{j}\\}_{j=1}^{\infty}$ in
$L^{2}(\mathbf{R}^{d})$. In what follows, let
$\operatorname{ON}_{0}=\operatorname{ON}_{0}(L^{2}(\mathbf{R}^{d}))$ be the
subset of $\operatorname{ON}(L^{2}(\mathbf{R}^{d}))$, which consists of all
_finite_ orthonormal sequences, $\\{f_{j}\\}_{j=1}^{N}$ in
$L^{2}(\mathbf{R}^{d})$ such that $f_{j}\in\Sigma_{1}(\mathbf{R}^{d})$ for
every $j\in\mathbf{Z}_{+}$. We also let $s_{0,A}(\mathbf{R}^{2d})$ be the set
of all $\mathfrak{a}\in\Sigma_{1}(\mathbf{R}^{2d})$ such that for some integer
$N\geq 1$, $\\{f_{j}\\}_{j=1}^{\infty}\in\operatorname{ON}_{0}$,
$\\{g_{j}\\}_{j=1}^{\infty}\in\operatorname{ON}_{0}$ and decreasing non-
negative sequence $\\{\lambda_{j}\\}_{j=1}^{\infty}$, we have

$\mathfrak{a}=\sum_{j=1}^{N}\lambda_{j}W_{f_{j},g_{j}}^{A}.$ (4.11)

It follows that $s_{0,A}(\mathbf{R}^{2d})$ is a vector space contained in
$\Sigma_{1}(\mathbf{R}^{2d})$.

A combination of Proposition 4.3 and spectral theorem (Proposition 4.1) gives
the first part of the following. Here recall that
$\check{\mathfrak{a}}(X)=\mathfrak{a}(-X)$.

######  Proposition 4.4.

Let $p\in\mathbf{R}_{\sharp}$, $A\in\mathbf{M}(d,\mathbf{R})$ and
$\mathfrak{a}\in s_{p}^{A}(\mathbf{R}^{2d})$. Then the following is true:

  1. (1)

if $p<\infty$, then there are
$\\{f_{j}\\}_{j=1}^{\infty}\in\operatorname{ON}(L^{2}(\mathbf{R}^{d}))$ and
$\\{g_{j}\\}_{j=1}^{\infty}\in\operatorname{ON}(L^{2}(\mathbf{R}^{d}))$ and a
unique sequence $\\{\lambda_{j}\\}_{j=1}^{\infty}\in\ell^{p}(\mathbf{Z}_{+})$
of decreasing non-negative real numbers such that

$\mathfrak{a}=\sum_{j=1}^{\infty}\lambda_{j}W_{f_{j},g_{j}}^{A}\quad\text{and}\quad\|\mathfrak{a}\|_{s_{p,A}}=(2\pi)^{-\frac{d}{2}}\|\\{\lambda_{j}\\}_{j=1}^{\infty}\|_{\ell^{p}}\text{;}$
(4.12)

  2. (2)

if $p<\infty$, then $s_{0,A}(\mathbf{R}^{2d})$ is dense in
$s_{p,A}(\mathbf{R}^{2d})$;

  3. (3)

if $X_{0}=(x_{0},\xi_{0})\in\mathbf{R}^{2d}$, then the mappings

$\mathfrak{a}\mapsto\mathfrak{a}(\,\cdot\,-X_{0})\quad\text{and}\quad\mathfrak{a}\mapsto\check{\mathfrak{a}}$
(4.13)

are bijective on $s_{0,A}(\mathbf{R}^{2d})$ and bijective isometries on
$s_{p,A}(\mathbf{R}^{2d})$.

For the proof it is convenient to have the following lemma. The result follows
by straight-forward computations and is left for the reader (see also [47,
Lemma 1.3]).

######  Lemma 4.5.

Let $A\in\mathbf{M}(\mathbf{R},d)$, and $T_{X_{0}}$ be the map on
$\Sigma_{1}^{\prime}(\mathbf{R}^{d})$, given by

$T_{X_{0}}f\equiv
f(\,\cdot\,-x_{0})e^{-i\langle\,\cdot\,,\xi_{0}\rangle},\qquad
f\in\Sigma_{1}^{\prime}(\mathbf{R}^{d}),$

when $X_{0}=(x_{0},\xi_{0})\in\mathbf{R}^{2d}$. Then

$W_{f,g}^{A}(X-X_{0})=W_{T_{X_{0}}f,T_{X_{0}}g}^{A}(X)\quad\text{and}\quad
W_{f,g}^{A}(-X)=W_{\check{f},\check{g}}(X),$

when $f,g\in\Sigma_{1}^{\prime}(\mathbf{R}^{d})$.

###### Proof of Proposition 4.4.

The assertion (1) follows from Proposition 4.1 and the fact that the map (4.6)
is isometric bijection between $s_{p,A}(\mathbf{R}^{2d})$ and
$\mathscr{I}_{p}(L^{2}(\mathbf{R}^{d}))$.

The assertion (2) follows from the same arguments and the fact that the set of
finite rank operators are dense in $\mathscr{I}_{p}(L^{2}(\mathbf{R}^{d}))$
when $p<\infty$.

In order to prove (3) we first observe that if $T_{X_{0}}$ is the same as in
Lemma 4.5, then the map

$\\{f_{j}\\}_{j=1}^{\infty}\mapsto\\{T_{X_{0}}f_{j}\\}_{j=1}^{\infty}$

is bijective on $\operatorname{ON}_{0}(L^{2}(\mathbf{R}^{d}))$ and on
$\operatorname{ON}(L^{2}(\mathbf{R}^{d}))$. The assertion now follows by
combining this fact with (1) and Lemma 4.5. ∎

######  Remark 4.6.

Suppose that $f,g\in L^{2}(\mathbf{R}^{d})$,
$p\in\mathbf{R}_{\sharp}\cup\\{0\\}$ and $A\in\mathbf{M}(d,\mathbf{R})$. Then
it follows by Fourier’s inversion formula that

$\|W_{f,g}^{A}\|_{L^{2}(\mathbf{R}^{2d})}=\|f\|_{L^{2}(\mathbf{R}^{d})}\|g\|_{L^{2}(\mathbf{R}^{d})},$

which is also equivalent with the so-called Moyal’s formula. Hence, by
previous proposition it follows that $W_{f,g}^{A}\in
s_{p,A}(\mathbf{R}^{2d})$, and

$\|W_{f,g}^{A}\|_{s_{p,A}}=(2\pi)^{-\frac{d}{2}}\|f\|_{L^{2}(\mathbf{R}^{d})}\|g\|_{L^{2}(\mathbf{R}^{d})}.$
(4.14)

######  Remark 4.7.

Let $A,B\in\mathbf{M}(d,\mathbf{R})$ and $p\in\mathbf{R}_{\sharp}\cup\\{0\\}$.
Then it follows from (4.7) that

$e^{i\langle AD_{\xi},D_{x}\rangle}s_{p,A}(\mathbf{R}^{2d})=e^{i\langle
BD_{\xi},D_{x}\rangle}s_{p,B}(\mathbf{R}^{2d}).$ (4.15)

If in addition $p>0$, then it also follows that the map
$e^{i\langle(A-B)D_{\xi},D_{x}\rangle}$ is an isometric bijection from
$s_{p,A}(\mathbf{R}^{2d})$ to $s_{p,B}(\mathbf{R}^{2d})$.

######  Remark 4.8.

Let $p\in[1,\infty]$, $r\in(0,1]$ and $A\in\mathbf{M}(\mathbf{R},d)$. Then
Proposition 4.4 shows that

$\displaystyle W\\!\\!L^{1,r}(\mathbf{R}^{2d})*s_{p,A}(\mathbf{R}^{2d})$
$\displaystyle\subseteq L^{1}(\mathbf{R}^{2d})*s_{p,A}(\mathbf{R}^{2d})$
$\displaystyle\subseteq\mathcal{M}(\mathbf{R}^{2d})*s_{p,A}(\mathbf{R}^{2d})=s_{p,A}(\mathbf{R}^{2d}).$
(4.16)

Here recall that $\mathcal{M}(\mathbf{R}^{2d})$ is the Banach space of
(complex-valued) measures on $\mathbf{R}^{2d}$ with bounded mass. (Cf. Example
1.9.)

In fact, the first two inclusions in (4.16) are obvious. By (4.13) it follows
that if $\mathfrak{a}$ is a simple function on $\mathbf{R}^{2d}$ and
$\mathfrak{b}\in s_{p,A}(\mathbf{R}^{2d})$, then $\mathfrak{a}*\mathfrak{b}\in
s_{p,A}(\mathbf{R}^{2d})$, and

$\|\mathfrak{a}*\mathfrak{b}\|_{s_{p,A}}\leq\|\mathfrak{a}\|_{L^{1}}\|\mathfrak{b}\|_{s_{p,A}},\quad\mathfrak{a}\in
L^{1}(\mathbf{R}^{2d}),\ \mathfrak{b}\in s_{p,A}(\mathbf{R}^{2d}).$ (4.17)

For general $\mathfrak{a}\in L^{1}(\mathbf{R}^{2d})$, the convolution
$\mathfrak{a}*\mathfrak{b}\in s_{p,A}(\mathbf{R}^{2d})$ and estimate (4.17)
now follow by continuous extensions, using the fact that the set of simple
functions are dense in $L^{1}(\mathbf{R}^{2d})$.

Since $\mathcal{M}(\mathbf{R}^{2d})$ is translation invariant, similar
arguments also show that for $\mathfrak{a}\in\mathcal{M}(\mathbf{R}^{2d})$ and
$\mathfrak{b}\in s_{p,A}(\mathbf{R}^{2d})$, then $\mathfrak{a}*\mathfrak{b}$
is uniquely defined as an element in $s_{p,A}(\mathbf{R}^{2d})$. That is,
$\mathcal{M}(\mathbf{R}^{2d})*s_{p,A}(\mathbf{R}^{2d})\subseteq
s_{p,A}(\mathbf{R}^{2d})$ Since $\mathcal{M}$ contains the unit element
$\delta_{0}$ for convolutions, it follows that equality is attained for the
last inclusion, giving (4.16).

We also observe that a combination of (4.16), (4.17) and Hahn-Banach’s theorem
give

$\|\mathfrak{a}*\mathfrak{b}\|_{s_{p,A}}\leq\|\mathfrak{a}\|_{\mathcal{M}}\|\mathfrak{b}\|_{s_{p,A}},\quad\mathfrak{a}\in\mathcal{M}(\mathbf{R}^{2d}),\
\mathfrak{b}\in s_{p,A}(\mathbf{R}^{2d}).$ (4.18)

Next we shall discuss compositions and other multiplications for Schatten-von
Neumann symbols. The following result concerns Hölder properties for such
symbol classes, and follows from Proposition 4.2. The details are left for the
reader.

######  Proposition 4.9.

Let $p,q,r\in\mathbf{R}_{\sharp}$ satisfy

$\frac{1}{p}+\frac{1}{q}=\frac{1}{r}\quad\text{and}\quad(p,q,r)\neq(\infty,\infty,\sharp),$

$A\in\mathbf{M}(d,\mathbf{R})$. Then the map
$(\mathfrak{a},\mathfrak{b})\mapsto\mathfrak{a}{\text{\footnotesize$\\#$}}_{A}\mathfrak{b}$
from $\Sigma_{1}(\mathbf{R}^{2d})\times\Sigma_{1}(\mathbf{R}^{2d})$ to
$\Sigma_{1}(\mathbf{R}^{2d})$ is uniquely extendable to a continuous map from
$s_{p,A}(\mathbf{R}^{2d})\times s_{q,A}(\mathbf{R}^{2d})$ to
$s_{r,A}(\mathbf{R}^{2d})$, and

$\|\mathfrak{a}{\text{\footnotesize$\\#$}}_{A}\mathfrak{b}\|_{s_{r,A}}\leq\|\mathfrak{a}\|_{s_{p,A}}\|\mathfrak{b}\|_{s_{q,A}},\qquad\mathfrak{a}\in
s_{p,A}(\mathbf{R}^{2d}),\ \mathfrak{b}\in s_{q,A}(\mathbf{R}^{2d}).$

###### Proof.

The result follows from the fact that
$\mathfrak{a}\mapsto\operatorname{Op}_{A}(\mathfrak{a})$ is an isometric
bijection from $s_{p,A}(\mathbf{R}^{2d})$ to $\mathscr{I}_{p}$, and that

$T_{1}\circ T_{2}\in\mathscr{I}_{r}\quad\text{and}\quad\|T_{1}\circ
T_{2}\|_{\mathscr{I}_{r}}\leq\|T_{1}\|_{\mathscr{I}_{p}}\|T_{2}\|_{\mathscr{I}_{q}}$

when $T_{1}\in\mathscr{I}_{p}$ and $T_{2}\in\mathscr{I}_{q}$. (See e. g.
[46].) ∎

###  4.3. The Weyl calculus, symplectic Fourier transform and twisted
convolution

Some properties in pseudo-differential operators take convenient forms in the
case of Weyl calculus, i. e. when $A=\frac{1}{2}\cdot I_{d}$. For convenience
we set

$\displaystyle\operatorname{Op}^{w}(\mathfrak{a})$
$\displaystyle=\operatorname{Op}_{A}(\mathfrak{a}),$ $\displaystyle\quad
s_{p,A}$ $\displaystyle=s_{p}^{w},\quad W_{f,g}=W_{f,g}^{A}$ (4.19)
$\displaystyle K_{\mathfrak{a}}^{w}$ $\displaystyle=K_{\mathfrak{a},A}$
$\displaystyle\quad\text{and}\qquad\mathfrak{a}{\text{\footnotesize$\\#$}}\mathfrak{b}$
$\displaystyle=\mathfrak{a}{\text{\footnotesize$\\#$}}_{A}\mathfrak{b},\quad\text{when}\quad
A={\textstyle{\frac{1}{2}}}\cdot I_{d},$

when $f,g\in\Sigma_{1}^{\prime}(\mathbf{R}^{d})$ and
$\mathfrak{a},\mathfrak{b}\in\Sigma_{1}^{\prime}(\mathbf{R}^{2d})$ are
suitable.

The Weyl quantization possess several properties which are violated for other
pseudo-differential calculi. For example, it is the only pseudo-differential
calculus which is so-called symplectic invariant, a fundamental property in
quantization.

There are several convenient links between the Weyl quantization, symplectic
Fourier transform and the twisted convolution (see (2.6) and (2.7)). For
example, beside (2.8) we have

$\displaystyle\mathfrak{a}{\text{\footnotesize$\\#$}}\mathfrak{b}$
$\displaystyle=(2\pi)^{-\frac{d}{2}}\mathfrak{a}*_{\sigma}(\mathscr{F}_{\sigma}\mathfrak{b})=(2\pi)^{-\frac{d}{2}}(\mathscr{F}_{\sigma}\mathfrak{a})*_{\sigma}\check{\mathfrak{b}},$
(4.20)
$\displaystyle\mathscr{F}_{\sigma}(\mathfrak{a}{\text{\footnotesize$\\#$}}\mathfrak{b})$
$\displaystyle=(2\pi)^{-\frac{d}{2}}(\mathscr{F}_{\sigma}\mathfrak{a})*_{\sigma}(\mathscr{F}_{\sigma}\mathfrak{b}),$
$\displaystyle K_{\mathscr{F}_{\sigma}\mathfrak{a}}^{w}$ $\displaystyle=S\circ
K_{\mathfrak{a}}^{w},$ $\displaystyle\quad(SK)(x,y)$ $\displaystyle=K(-x,y)$
(4.21) and $\displaystyle\mathscr{F}_{\sigma}W_{f,g}$
$\displaystyle=W_{\check{f},g},$ $\displaystyle\quad\check{f}(x)$
$\displaystyle=f(x),$ (4.22)

when $\mathfrak{a},\mathfrak{b}\in\mathcal{S}_{s}^{\prime}(\mathbf{R}^{2d})$
are suitable and $f,g\in\mathcal{S}_{s}^{\prime}(\mathbf{R}^{d})$ (cf.
(4.19)). We also have that

$\mathscr{F}_{\sigma}:s_{p}^{w}(\mathbf{R}^{2d})\to
s_{p}^{w}(\mathbf{R}^{2d})$ (4.23)

is an isometric bijection for every $p\in\mathbf{R}_{\sharp}\cup\\{0\\}$.

A combination of Propositions 4.4 (3) and 4.9, (4.20) and (4.23) gives the
following. The details are left for the reader.

######  Proposition 4.10.

Let $p$, $q$ and $r$ be as in Proposition 4.9. Then the map
$(\mathfrak{a},\mathfrak{b})\mapsto\mathfrak{a}*_{\sigma}\mathfrak{b}$ is
continuous from $s_{p}^{w}(\mathbf{R}^{2d})\times s_{q}^{w}(\mathbf{R}^{2d})$
to $s_{r}^{w}(\mathbf{R}^{2d})$, and

$\|\mathfrak{a}*_{\sigma}\mathfrak{b}\|_{s_{r}^{w}}\leq(2\pi)^{-\frac{d}{2}}\|\mathfrak{a}\|_{s_{p}^{w}}\|\mathfrak{b}\|_{s_{q}^{w}},\qquad\mathfrak{a}\in
s_{p}^{w}(\mathbf{R}^{2d}),\ \mathfrak{b}\in s_{q}^{w}(\mathbf{R}^{2d}).$

For future references we also recall that

$\displaystyle W\\!\\!L^{1,r}(\mathbf{R}^{2d})\subseteq
L^{1}(\mathbf{R}^{2d})$ $\displaystyle\subseteq
s_{\sharp}^{w}(\mathbf{R}^{2d})\cap\mathcal{M}(\mathbf{R}^{2d})$ (4.24)
$\displaystyle\subseteq
s_{\sharp}^{w}(\mathbf{R}^{2d})\cup\mathcal{M}(\mathbf{R}^{2d})\subseteq
s_{\infty}^{w}(\mathbf{R}^{2d}),\quad r\in(0,1],$

with continuous inclusions. (See e. g. [32].)

###  4.4. Convolutions of Schatten-von Neumann symbols

For convolutions we have the following.

######  Proposition 4.11.

Suppose $p,q,r\in(0,\infty]$ and $A\in\mathbf{M}(d,\mathbf{R})$ satisfy

$\frac{1}{p}+\frac{1}{q}=1+\frac{1}{r},\qquad p,q,r\geq 1,$

or $q\leq p\leq r\leq 1$. Then the map
$(\mathfrak{a},\mathfrak{b})\mapsto\mathfrak{a}*\mathfrak{b}$ from
$\Sigma_{1}(\mathbf{R}^{2d})\times\Sigma_{1}(\mathbf{R}^{2d})$ to
$\Sigma_{1}(\mathbf{R}^{2d})$ is uniquely extendable to a continuous map from

$W\\!\\!L^{\max(q,1),q}(\mathbf{R}^{2d})\times
s_{p,A}(\mathbf{R}^{2d})\quad\text{or}\quad s_{p,A}(\mathbf{R}^{2d})\times
W\\!\\!L^{\max(q,1),q}(\mathbf{R}^{2d})$

to $s_{r,A}(\mathbf{R}^{2d})$.

###### Proof.

By (4.15) and that

$e^{i\langle AD_{\xi},D_{x}\rangle}(\mathfrak{a}*\mathfrak{b})=(e^{i\langle
AD_{\xi},D_{x}\rangle}\mathfrak{a})*\mathfrak{b},$ (4.25)

we reduce ourself to the Weyl case $A=\frac{1}{2}I_{d}$. For $p,q,r\geq 1$,
the result follows from [47, 56].

We need to consider the case when $q\leq p\leq r\leq 1$. Then the facts that
$s_{p,A}(\mathbf{R}^{2d})$ and $W\\!\\!L^{1,q}(\mathbf{R}^{2d})$ increase with
$p$ and $q$, respectively, reduce ourselves to the case when $p=q=r\leq 1$.

First we assume that $\mathfrak{a}=W_{f,f}^{A}=W_{f,f}$ with $f\in
L^{2}(\mathbf{R}^{d})$, and that $0\leq\mathfrak{b}\in
W\\!\\!L^{1,r}(\mathbf{R}^{2d})$. Then $\mathfrak{a}$ is continuous and real-
valued. Let $Q=[0,1]^{2d}$. The mean-value theorem gives that for some
$Z_{j}\in Q$, $j\in\mathbf{Z}^{d}$,

$\displaystyle\|\mathfrak{a}*\mathfrak{b}\|_{s_{r}^{w}}^{r}$
$\displaystyle=\left\|\sum_{j\in\mathbf{Z}^{d}}\int_{j+Q}\mathfrak{a}(\,\cdot\,-Y)\mathfrak{b}(Y)\,dy\right\|_{s_{r}^{w}}^{r}$
$\displaystyle=\left\|\sum_{j\in\mathbf{Z}^{d}}\mathfrak{a}(\,\cdot\,-j-Z_{j})\int_{j+Q}\mathfrak{b}(Y)\,dy\right\|_{s_{r}^{w}}^{r}$
$\displaystyle\leq\sum_{j\in\mathbf{Z}^{d}}\|\mathfrak{a}(\,\cdot\,-j-Z_{j})\|_{s_{r}^{w}}^{r}\left(\int_{j+Q}\mathfrak{b}(Y)\,dy\right)^{r}$
$\displaystyle=\sum_{j\in\mathbf{Z}^{d}}\|\mathfrak{a}\|_{s_{r}^{w}}^{r}\left(\int_{j+Q}\mathfrak{b}(Y)\,dy\right)^{r}=\|\mathfrak{a}\|_{s_{r}^{w}}^{r}\|\mathfrak{b}\|_{W\\!\\!L^{1,r}}^{r},$

and the result follows in this case.

Next assume that $\mathfrak{a}=W_{f,f}$ as before but $\mathfrak{b}\in
W\\!\\!L^{1,r}(\mathbf{R}^{2d})$ being general. By splitting up $\mathfrak{b}$
in positive and negative real and imaginary parts

$\mathfrak{b}=\mathfrak{b}_{1}-\mathfrak{b}_{2}+i(\mathfrak{b}_{3}-\mathfrak{b}_{4}),$

with

$\displaystyle\mathfrak{b}_{1}$
$\displaystyle=\max(\operatorname{Re}(\mathfrak{b}),0),$
$\displaystyle\quad\mathfrak{b}_{2}$
$\displaystyle=\max(-\operatorname{Re}(\mathfrak{b}),0),$
$\displaystyle\mathfrak{b}_{3}$
$\displaystyle=\max(\operatorname{Im}(\mathfrak{b}),0)$
$\displaystyle\quad\text{and}\quad\mathfrak{b}_{4}$
$\displaystyle=\max(-\operatorname{Im}(\mathfrak{b}),0),$

the first part of the proof gives

$\displaystyle\|\mathfrak{a}*\mathfrak{b}\|_{s_{r}^{w}}^{r}$
$\displaystyle\leq\|\mathfrak{a}*\mathfrak{b}_{1}\|_{s_{r}^{w}}^{r}+\|\mathfrak{a}*\mathfrak{b}_{2}\|_{s_{r}^{w}}^{r}+\|\mathfrak{a}*\mathfrak{b}_{3}\|_{s_{r}^{w}}^{r}+\|\mathfrak{a}*\mathfrak{b}_{4}\|_{s_{r}^{w}}^{r}$
$\displaystyle\leq\|\mathfrak{a}\|_{s_{r}^{w}}^{r}\sum_{k=1}^{4}\|\mathfrak{b}_{k}\|_{W\\!\\!L^{1,r}}^{r}\leq
4\|\mathfrak{a}\|_{s_{r}^{w}}^{r}\|\mathfrak{b}\|_{W\\!\\!L^{1,r}}^{r},$

and the result follows in this case as well.

Next suppose that $\mathfrak{a}=W_{f,g}$ with $\|f\|_{L^{2}}=\|g\|_{L^{2}}=1$
and that $\mathfrak{b}\in W\\!\\!L^{1,r}(\mathbf{R}^{2d})$. Then

$\mathfrak{a}=\frac{1}{4}(\mathfrak{a}_{1}-\mathfrak{a}_{2}+i(\mathfrak{a}_{3}-\mathfrak{a}_{4})),$

where

$\displaystyle\mathfrak{a}_{k}$
$\displaystyle=W_{f+\theta_{k}g,f+\theta_{k}g},$
$\displaystyle\quad\theta_{1}=1,\ \theta_{2}=-1,\
\theta_{3}=i\quad\text{and}\quad\theta_{4}=-i.$

We have

$\displaystyle\|\mathfrak{a}_{k}\|_{s_{r}^{w}}$
$\displaystyle=\|W_{f+\theta_{k}g,f+\theta_{k}g}\|_{s_{r}^{w}}=(2\pi)^{\frac{d}{2}}\|f+\theta_{k}g\|_{L^{2}}^{2}$
$\displaystyle\leq(2\pi)^{\frac{d}{2}}(\|f\|_{L^{2}}+\|g\|_{L^{2}})^{2}=4(2\pi)^{\frac{d}{2}}=4\|\mathfrak{a}\|_{s_{r}^{w}}.$

By the previous case we get

$\displaystyle\|\mathfrak{a}*\mathfrak{b}\|_{s_{r}^{w}}^{r}$
$\displaystyle\leq
4^{-r}\sum_{k=1}^{4}\|\mathfrak{a}_{k}*\mathfrak{b}\|_{s_{r}^{w}}^{r}$
$\displaystyle\leq
4^{-r}\sum_{k=1}^{4}4\|\mathfrak{a}_{k}\|_{s_{r}^{w}}^{r}\|\mathfrak{b}\|_{W\\!\\!L^{1,r}}^{r}$
$\displaystyle\leq
16\|\mathfrak{a}\|_{s_{r}^{w}}^{r}\|\mathfrak{b}\|_{W\\!\\!L^{1,r}}^{r},$

and the result follows in this case as well.

Finally, for general $\mathfrak{a}$ given by (4.12) with $p=r$, we get

$\displaystyle\|\mathfrak{a}*\mathfrak{b}\|_{s_{r}^{w}}^{r}$
$\displaystyle=\left\|\sum_{j=1}^{\infty}\lambda_{j}W_{f_{j},g_{j}}*\mathfrak{b}\right\|_{s_{r}^{w}}^{r}\leq\sum_{j=1}^{\infty}\lambda_{j}^{r}\|W_{f_{j},g_{j}}*\mathfrak{b}\|_{s_{r}^{w}}^{r}$
$\displaystyle\lesssim\left(\sum_{j=1}^{\infty}\lambda_{j}^{r}\right)\|\mathfrak{b}\|_{W\\!\\!L^{1,r}}^{r}\asymp\|\mathfrak{a}\|_{s_{r}^{w}}^{r}\|\mathfrak{b}\|_{W\\!\\!L^{1,r}}^{r},$

and the result follows. ∎

###  4.5. Convolution factorizations of Schatten-von Neumann symbols

The following result shows that any Schatten-von Neumann symbol can be
factorized through convolutions into elements in the same class and elements
in the Wiener amalgam space $W\\!\\!L^{1,r}(\mathbf{R}^{2d})$, for suitable
$r\in(0,1]$.

######  Theorem 4.12.

Suppose $r\in(0,1]$, $p\in[r,\sharp]$ and $A\in\mathbf{M}(d,\mathbf{R})$. Then

$W\\!\\!L^{1,r}(\mathbf{R}^{2d})*s_{p,A}(\mathbf{R}^{2d})=s_{p,A}(\mathbf{R}^{2d})*W\\!\\!L^{1,r}(\mathbf{R}^{2d})=s_{p,A}(\mathbf{R}^{2d}).$
(4.26)

###### Proof.

By (4.15) and (4.25), we reduce ourself to the Weyl case $A=\frac{1}{2}I_{d}$.
In particular it follows that

$W_{f,f}(x,\xi)=W_{f,f}^{A}(x,\xi)\in\mathbf{R}$

when $f\in L^{2}(\mathbf{R}^{d})$ and $x,\xi\in\mathbf{R}^{d}$. By the
commutativity of the convolution product it suffices to prove

$s_{p,A}(\mathbf{R}^{2d})*W\\!\\!L^{1,r}(\mathbf{R}^{2d})=s_{p,A}(\mathbf{R}^{2d}).$

Let $\phi\in C_{0}^{\infty}(\mathbf{R}^{2d};[0,1])$ be such that

$\int_{\mathbf{R}^{2d}}\phi(Y)\,dY=1\quad\text{and}\quad\operatorname{supp}(\phi)\subseteq
Q_{M}\equiv[-1,1]^{2d},$

and let $\phi_{\varepsilon}(X)=\varepsilon^{-2d}\phi(\varepsilon^{-1}X)$ for
any $\varepsilon>0$, $X=(x,\xi)\in\mathbf{R}^{2d}$. It follows by straight-
forward computations that if $f,g\in L^{2}(\mathbf{R}^{d})$, then

$(W_{f,g}*\phi_{\varepsilon})(x,\xi)=\int_{\mathbf{R}^{2d}}W_{f_{\varepsilon
Y},g_{\varepsilon Y}}(x,\xi)\phi(Y)\,dY,$

where

$f_{Y}(x)=f(x-y)e^{-i\langle x,\eta\rangle},\quad g_{Y}(x)=g(x-y)e^{-i\langle
x,\eta\rangle},\quad Y=(y,\eta)\in\mathbf{R}^{2d}.$

Hence, if $\mathfrak{a}$ is expressed as in (4.12), then

$\displaystyle\mathfrak{a}*\phi_{\varepsilon}-\mathfrak{a}$
$\displaystyle=\mathfrak{b}_{\varepsilon}+\mathfrak{c}_{\varepsilon}$ (4.27)
where $\displaystyle\mathfrak{b}_{\varepsilon}$
$\displaystyle=\int_{Q_{M}}\left(\sum_{j=1}^{\infty}\lambda_{j}W_{f_{j,\varepsilon
Y},g_{j,\varepsilon Y}-g_{j}}\right)\phi(Y)\,dY$ (4.28)
$\displaystyle=\sum_{j=1}^{\infty}\lambda_{j}W_{f_{j,\varepsilon
Y_{0}},g_{j,\varepsilon Y_{0}}-g_{j}}$ (4.29) and
$\displaystyle\mathfrak{c}_{\varepsilon}$
$\displaystyle=\int_{\mathbf{R}^{2d}}\left(\sum_{j=1}^{\infty}\lambda_{j}W_{f_{j,\varepsilon
Y}-f_{j},g_{j}}\right)\phi(Y)\,dY.$ (4.30)

Since $\|\phi_{\varepsilon}\|_{W\\!\\!L^{1,r}}\leq C$ for some constant $C>0$
which is independent of $\varepsilon\in(0,1]$, the assertion will follow if we
prove

$\lim_{\varepsilon\to
0+}\|\mathfrak{b}_{\varepsilon}\|_{s_{p}^{w}}=0\quad\text{and}\quad\lim_{\varepsilon\to
0+}\|\mathfrak{c}_{\varepsilon}\|_{s_{p}^{w}}=0,$ (4.31)

in view of Theorem 1.5.

We have

$\mathfrak{a}-\sum_{j=1}^{N}\lambda_{j}W_{f_{j},g_{j}}\to 0$

as $N\to\infty$ with convergence in $s_{p,A}(\mathbf{R}^{2d})$. By combining
this with Proposition 4.11, it follows that

$\left(\mathfrak{a}-\sum_{j=1}^{N}\lambda_{j}W_{f_{j},g_{j}}\right)*\phi\to 0$

as $N\to\infty$, again with convergence in $s_{p}^{w}(\mathbf{R}^{2d})$. From
these limits we reduce ourself to the case where we may assume that

$\mathfrak{a}=\sum_{j=1}^{N}\lambda_{j}W_{f_{j},g_{j}},$ (4.32)

for some integer $N\geq 0$.

Our aim is to apply the mean-value theorem and Minkowski’s inequality to reach
desired estimates. We then need to reformulate $\mathfrak{b}_{\varepsilon}$ in
suitable ways. We have

$\displaystyle\mathfrak{b}_{\varepsilon}$
$\displaystyle=\frac{1}{4}\sum_{j=1}^{N}\lambda_{j}\mathfrak{b}_{j,\varepsilon},\qquad\mathfrak{b}_{j,\varepsilon}=\mathfrak{b}_{1,j,\varepsilon}+i\mathfrak{b}_{2,j,\varepsilon}-\mathfrak{b}_{3,j,\varepsilon}-i\mathfrak{b}_{4,j,\varepsilon},$
(4.33) where $\displaystyle\mathfrak{b}_{k,j,\varepsilon}$
$\displaystyle=\int_{Q_{M}}W_{h_{k,j,\varepsilon Y},h_{k,j,\varepsilon
Y}}\phi(Y)\,dY,\qquad h_{k,j,Y}=f_{j,Y}+i^{k-1}(g_{j,Y}-g),$ (4.34)

$k=1,2,3,4$. Since $\phi\geq 0$ satisfies

$\int_{\mathbf{R}^{2d}}\phi(Y)\,dY=1$

and $W_{h_{k,j,\varepsilon Y},h_{k,j,\varepsilon Y}}$ is real-valued and
smooth, the mean value theorem gives

$\mathfrak{b}_{k,j,\varepsilon}=W_{h_{k,j,\varepsilon
Y_{k,j}},h_{k,j,\varepsilon Y_{k,j}}},$

for some $Y_{k,j}=Y_{k,j,\varepsilon}\in Q_{M}$.

By rearranging terms we obtain

$\displaystyle\mathfrak{b}_{j,\varepsilon}$
$\displaystyle=\mathfrak{b}^{0}_{1,j,\varepsilon}+i\mathfrak{b}^{0}_{2,j,\varepsilon}+\mathfrak{b}^{0}_{3,j,\varepsilon}+i\mathfrak{b}^{0}_{4,j,\varepsilon}$
(4.35) where $\displaystyle\mathfrak{b}^{0}_{1,j,\varepsilon}$
$\displaystyle=4W_{f_{j,\varepsilon Y_{1,j}},g_{j,\varepsilon
Y_{1,j}}-g_{j}},$ (4.36) $\displaystyle\mathfrak{b}^{0}_{2,j,\varepsilon}$
$\displaystyle=-W_{f_{j,\varepsilon Y_{1,j}}-f_{j,\varepsilon
Y_{2,j}},f_{j,\varepsilon Y_{1,j}}+ig_{j,\varepsilon
Y_{1,j}}}-iW_{g_{j,\varepsilon Y_{1,j}}-g_{j,\varepsilon
Y_{2,j}},f_{j,\varepsilon Y_{1,j}}+ig_{j,\varepsilon Y_{1,j}}}$
$\displaystyle+iW_{f_{j,\varepsilon Y_{2,j}}+ig_{j,\varepsilon
Y_{2,j}},f_{j,\varepsilon Y_{1,j}}-f_{j,\varepsilon
Y_{2,j}}}+W_{f_{j,\varepsilon Y_{2,j}}+ig_{j,\varepsilon
Y_{2,j}},g_{j,\varepsilon Y_{1,j}}-g_{j,\varepsilon Y_{2,j}}}$
$\displaystyle+2\operatorname{Im}\left(W_{f_{j,\varepsilon
Y_{1,j}}-f_{j,\varepsilon Y_{2,j}},g_{j}}-iW_{g_{j,\varepsilon
Y_{1,j}}-g_{j,\varepsilon Y_{2,j}},g_{j}}\right)$ (4.37)
$\displaystyle\mathfrak{b}^{0}_{3,j,\varepsilon}$
$\displaystyle=W_{f_{j,\varepsilon Y_{1,j}}-f_{j,\varepsilon
Y_{3,j}},f_{j,\varepsilon Y_{1,j}}-g_{j,\varepsilon
Y_{1,j}}}-W_{g_{j,\varepsilon Y_{1,j}}-g_{j,\varepsilon
Y_{3,j}},f_{j,\varepsilon Y_{1,j}}-g_{j,\varepsilon Y_{1,j}}}$
$\displaystyle+W_{f_{j,\varepsilon Y_{3,j}}-g_{j,\varepsilon
Y_{3,j}},f_{j,\varepsilon Y_{1,j}}-f_{j,\varepsilon
Y_{3,j}}}-W_{f_{j,\varepsilon Y_{3,j}}-g_{j,\varepsilon
Y_{3,j}},g_{j,\varepsilon Y_{1,j}}-g_{j,\varepsilon Y_{3,j}}}$
$\displaystyle+2\operatorname{Re}\left(W_{f_{j,\varepsilon
Y_{1,j}}-f_{j,\varepsilon Y_{3,j}},g_{j}}-W_{g_{j,\varepsilon
Y_{1,j}}-g_{j,\varepsilon Y_{3,j}},g_{j}}\right)$ (4.38) and
$\displaystyle\mathfrak{b}^{0}_{4,j,\varepsilon}$
$\displaystyle=W_{f_{j,\varepsilon Y_{1,j}}-f_{j,\varepsilon
Y_{4,j}},f_{j,\varepsilon Y_{1,j}}-ig_{j,\varepsilon
Y_{1,j}}}-iW_{g_{j,\varepsilon Y_{1,j}}-g_{j,\varepsilon
Y_{4,j}},f_{j,\varepsilon Y_{1,j}}-ig_{j,\varepsilon Y_{1,j}}}$
$\displaystyle+W_{f_{j,\varepsilon Y_{4,j}}-ig_{j,\varepsilon
Y_{4,j}},f_{j,\varepsilon Y_{1,j}}-f_{j,\varepsilon
Y_{4,j}}}+iW_{f_{j,\varepsilon Y_{4,j}}-ig_{j,\varepsilon
Y_{4,j}},g_{j,\varepsilon Y_{1,j}}-g_{j,\varepsilon Y_{4,j}}}$
$\displaystyle+2\operatorname{Im}\left(W_{f_{j,\varepsilon
Y_{1,j}}-f_{j,\varepsilon Y_{4,j}},g_{j}}-iW_{g_{j,\varepsilon
Y_{1,j}}-g_{j,\varepsilon Y_{4,j}},g_{j}}\right).$ (4.39)

Since $\overline{W_{f,g}(x,\xi)}=W_{g,f}(x,\xi)$, giving that

$2\operatorname{Re}(W_{f,g})=W_{f,g}+W_{g,f}\quad\text{and}\quad
2\operatorname{Im}(W_{f,g})=i^{-1}(W_{f,g}-W_{g,f}),$

it follows that all terms in (4.36)–(4.39) are of the forms

$\displaystyle W_{h_{1,j},g_{j,\varepsilon Y_{k,j}}-g_{j}},$ $\displaystyle
W_{f_{j,\varepsilon Y_{k,j}}-f_{j,\varepsilon Y_{m,j}},h_{2,j}},$
$\displaystyle W_{g_{j,\varepsilon Y_{k,j}}-g_{j,\varepsilon
Y_{m,j}},h_{3,j}},$ (4.40) $\displaystyle W_{h_{4,j},f_{j,\varepsilon
Y_{k,j}}-f_{j,\varepsilon Y_{m,j}}}$ and $\displaystyle
W_{h_{5,j},g_{j,\varepsilon Y_{k,j}}-g_{j,\varepsilon Y_{m,j}}},$

for some $h_{k,j}\in\Sigma_{1}(\mathbf{R}^{d})$, $k=1,\dots,5$. Since
$\Sigma_{1}(\mathbf{R}^{d})$ is dense in $L^{2}(\mathbf{R}^{d})$, a straight-
forward application of Lebesgue’s theorem gives

$\lim_{\varepsilon\to 0+}\|f_{j,\varepsilon
Y_{k,j}}-f_{j}\|_{L^{2}}=\lim_{\varepsilon\to 0+}\|g_{j,\varepsilon
Y_{k,j}}-g_{j}\|_{L^{2}}=0,$

which in turn leads to

$\lim_{\varepsilon\to 0+}\|f_{j,\varepsilon Y_{k,j}}-f_{j,\varepsilon
Y_{m,j}}\|_{L^{2}}=\lim_{\varepsilon\to 0+}\|g_{j,\varepsilon
Y_{k,j}}-g_{j,\varepsilon Y_{m,j}}\|_{L^{2}}=0.$

An application of (4.14) now gives

$\lim_{\varepsilon\to 0+}\|W_{f_{j,\varepsilon Y_{k,j}}-f_{j,\varepsilon
Y_{m,j}},h_{2,j}}\|_{s_{r}^{w}}\\\\[4.30554pt] =\lim_{\varepsilon\to
0+}\left((2\pi)^{-\frac{d}{2}}\|f_{j,\varepsilon Y_{k,j}}-f_{j,\varepsilon
Y_{m,j}}\|_{L^{2}}\|h_{2,j}\|_{L^{2}}\right)=0$

for the second expression in (4.40). In the same way we obtain

$\displaystyle\lim_{\varepsilon\to 0+}\|W_{h_{1,j},g_{j,\varepsilon
Y_{k,j}}-g_{j}}\|_{s_{r}^{w}}$ $\displaystyle=\lim_{\varepsilon\to
0+}\|W_{g_{j,\varepsilon Y_{k,j}}-g_{j,\varepsilon
Y_{m,j}},h_{3,j}}\|_{s_{r}^{w}}$ $\displaystyle=\lim_{\varepsilon\to
0+}\|W_{h_{4,j},f_{j,\varepsilon Y_{k,j}}-f_{j,\varepsilon
Y_{m,j}}}\|_{s_{r}^{w}}$ $\displaystyle=\lim_{\varepsilon\to
0+}\|W_{h_{5,j},g_{j,\varepsilon Y_{k,j}}-g_{j,\varepsilon
Y_{m,j}}}\|_{s_{r}^{w}}=0$

for the other expressions in (4.40).

Since $\mathfrak{b}^{0}_{k,j,\varepsilon}$ are linear combinations of Wigner
distributions in (4.40), it now follows from the previous estimates and the
quasi-norm properties for $\|\,\cdot\,\|_{s_{r}^{w}}$ that

$\lim_{\varepsilon\to
0+}\|\mathfrak{b}^{0}_{k,j,\varepsilon}\|_{s_{r}^{w}}=0,\quad k=1,2,3,4,$

which leads to

$0\leq\lim_{\varepsilon\to
0+}\|\mathfrak{b}^{0}_{j,\varepsilon}\|_{s_{p}^{w}}\leq\lim_{\varepsilon\to
0+}\|\mathfrak{b}^{0}_{j,\varepsilon}\|_{s_{r}^{w}}=0.$

The first limit in (4.31) now follows from the previous limit and (4.33).

By similar arguments, the second limit in (4.31) follows. The details are left
for the reader, and the result follows. ∎

###  4.6. Symbol product and twisted convolution factorizations of Schatten-
von Neumann symbols

The following result shows among others that any Schatten-von Neumann symbol
in the Weyl calculus can be factorized through twisted convolutions or Weyl
products into elements in the same class and elements in the Wiener amalgam
space $W\\!\\!L^{1,r}(\mathbf{R}^{2d})$, for suitable $r\in(0,1]$.

######  Theorem 4.13.

Suppose $p\in(0,\sharp]$ and $r\in(0,1]$. Then

$\displaystyle s_{p}^{w}(\mathbf{R}^{2d})$
$\displaystyle*_{\sigma}W\\!\\!L^{1,r}(\mathbf{R}^{2d})=s_{p}^{w}(\mathbf{R}^{2d})*_{\sigma}L^{1}(\mathbf{R}^{2d})$
$\displaystyle=s_{p}^{w}(\mathbf{R}^{2d})*_{\sigma}s_{\sharp}^{w}(\mathbf{R}^{2d})=s_{p}^{w}(\mathbf{R}^{2d})*_{\sigma}s_{\infty}^{w}(\mathbf{R}^{2d})=s_{p}^{w}(\mathbf{R}^{2d})$
(4.41) and $\displaystyle s_{p}^{w}(\mathbf{R}^{2d})$
$\displaystyle{\text{\footnotesize$\\#$}}W\\!\\!L^{1,r}(\mathbf{R}^{2d})=s_{p}^{w}(\mathbf{R}^{2d}){\text{\footnotesize$\\#$}}L^{1}(\mathbf{R}^{2d})$
$\displaystyle=s_{p}^{w}(\mathbf{R}^{2d}){\text{\footnotesize$\\#$}}s_{\sharp}^{w}(\mathbf{R}^{2d})=s_{p}^{w}(\mathbf{R}^{2d}){\text{\footnotesize$\\#$}}s_{\infty}^{w}(\mathbf{R}^{2d})=s_{p}^{w}(\mathbf{R}^{2d}).$
(4.42)

###### Proof.

By (4.20) and (4.23), it follows that (4.42) is equivalent to (4.41). Hence it
suffices to prove (4.41). By Proposition 4.10 it follows that

$s_{p}^{w}(\mathbf{R}^{2d})*_{\sigma}s_{\infty}^{w}(\mathbf{R}^{2d})\subseteq
s_{p}^{w}(\mathbf{R}^{2d}).$

Hence it suffices to prove

$s_{p}^{w}(\mathbf{R}^{2d})*_{\sigma}W\\!\\!L^{1,r}(\mathbf{R}^{2d})=s_{p}^{w}(\mathbf{R}^{2d})$
(4.43)

in view of (4.24).

Let

$\displaystyle\phi(x,\xi)$
$\displaystyle=\left(\frac{\pi}{2}\right)^{\frac{d}{2}}\psi(x)\psi(\xi),$
where $\displaystyle\psi$
$\displaystyle\in\Sigma_{1}(\mathbf{R}^{d}),\quad\int_{\mathbf{R}^{d}}\psi(x)\,dx=1$
(4.44)

and set $\phi_{\varepsilon}=\varepsilon^{-2d}\phi(\varepsilon^{-1}\,\cdot\,)$.
Since $\|\phi_{\varepsilon}\|_{W\\!\\!L^{1,r}}\leq C$ for some constant $C>0$
which is independent of $\varepsilon\in(0,1]$, the result follows from Theorem
1.5 if we prove that

$\lim_{\varepsilon\to
0+}\|\phi_{\varepsilon}*_{\sigma}\mathfrak{a}-\mathfrak{a}\|_{s_{p}^{w}}=0,$
(4.45)

for every $\mathfrak{a}\in s_{p}^{w}(\mathbf{R}^{2d})$.

By Proposition 4.9 and (4.24) it follows that the map
$(\mathfrak{a},\mathfrak{b})\mapsto\mathfrak{a}*_{\sigma}\mathfrak{b}$ is
continuous from $L^{1}(\mathbf{R}^{2d})\times s_{p}^{w}(\mathbf{R}^{2d})$ to
$s_{p}^{w}(\mathbf{R}^{2d})$. Hence it suffices to prove (4.45) when
$\mathfrak{a}\in s_{0}^{w}(\mathbf{R}^{2d})$, in view of Proposition 4.4. That
is, we may assume that $\mathfrak{a}$ is given by (4.32) for some integer
$N\geq 1$, $\\{f_{j}\\}_{j=1}^{\infty}\in\operatorname{ON}_{0}$,
$\\{g_{j}\\}_{j=1}^{\infty}\in\operatorname{ON}_{0}$ and
$\lambda_{j}\geq\lambda_{j+1}\geq 0$, $j=1,\dots,N-1$.

Let $r=\min(1,p)$. By triangle inequality we get

$\displaystyle\|\phi_{\varepsilon}*_{\sigma}\mathfrak{a}-\mathfrak{a}\|_{s_{p}^{w}}^{r}$
$\displaystyle=\left\|\sum_{j=1}^{N}\lambda_{j}\big{(}\phi_{\varepsilon}*_{\sigma}W_{f_{j},g_{j}}-W_{f_{j},g_{j}}\big{)}\right\|_{s_{p}^{w}}^{r}$
$\displaystyle\leq\sum_{j=1}^{N}\lambda_{j}^{r}\|\phi_{\varepsilon}*_{\sigma}W_{f_{j},g_{j}}-W_{f_{j},g_{j}}\|_{s_{p}^{w}}^{r}$
$\displaystyle\leq\sum_{j=1}^{N}\lambda_{j}^{r}\|\phi_{\varepsilon}*_{\sigma}W_{f_{j},g_{j}}-W_{f_{j},g_{j}}\|_{s_{r}^{w}}^{r}.$
(4.46)

Since

$\displaystyle\phi_{\varepsilon}*_{\sigma}W_{f_{j},g_{j}}$
$\displaystyle=(2\pi)^{\frac{d}{2}}\phi_{\varepsilon}{\text{\footnotesize$\\#$}}(\mathscr{F}_{\sigma}W_{f_{j},g_{j}})$
$\displaystyle=(2\pi)^{\frac{d}{2}}\phi_{\varepsilon}{\text{\footnotesize$\\#$}}W_{\check{f}_{j},g_{j}}=(2\pi)^{\frac{d}{2}}W_{\operatorname{Op}^{w}(\phi_{\varepsilon})\check{f}_{j},g_{j}},$

in view of Proposition 4.3 and (4.20), we get

$\displaystyle\|\phi_{\varepsilon}*_{\sigma}W_{f_{j},g_{j}}-W_{f_{j},g_{j}}\|_{s_{r}^{w}}$
$\displaystyle=\|W_{h_{\varepsilon,j},g_{j}}\|_{s_{r}^{w}}$ (4.47)
$\displaystyle=(2\pi)^{-\frac{d}{2}}\|h_{\varepsilon,j}\|_{L^{2}}\|g_{j}\|_{L^{2}}=(2\pi)^{-\frac{d}{2}}\|h_{\varepsilon,j}\|_{L^{2}},$

where

$h_{\varepsilon,j}=(2\pi)^{\frac{d}{2}}\operatorname{Op}^{w}(\phi_{\varepsilon})\check{f}_{j}-f_{j}.$

Hence, in view of (4.46) and (4.47), the result will follow if we prove

$\lim_{\varepsilon\to 0+}\|h_{\varepsilon,j}\|_{L^{2}}=0.$ (4.48)

By (4.44) we have

$\displaystyle h_{\varepsilon,j}(x)$
$\displaystyle=2^{-d}\varepsilon^{-2d}\iint_{\mathbf{R}^{2d}}\psi((2\varepsilon)^{-1}(x+y))\psi(\varepsilon^{-1}\xi)f_{j}(-y)e^{i\langle
x-y,\xi\rangle}\,dyd\xi-f_{j}(x)$
$\displaystyle=2^{-d}\varepsilon^{-2d}\iint_{\mathbf{R}^{2d}}\psi((2\varepsilon)^{-1}y)\psi(\varepsilon^{-1}\xi)f_{j}(x-y)e^{i\langle
x-y,\xi\rangle}\,dyd\xi-f_{j}(x)$
$\displaystyle=\iint_{\mathbf{R}^{2d}}\psi(y)\psi(\xi)f_{j}(x-2\varepsilon
y)e^{i\langle x-2\varepsilon y,\xi\rangle}\,dyd\xi-f_{j}(x)$
$\displaystyle=\iint_{\mathbf{R}^{2d}}\psi(y)\psi(\xi)f_{j}(x-2\varepsilon
y)e^{i\langle x-2\varepsilon y,\xi\rangle}\,dyd\xi-f_{j}(x)$
$\displaystyle=\int_{\mathbf{R}^{d}}\psi(y)((2\pi)^{\frac{d}{2}}\widehat{\psi}(\varepsilon
x-2\varepsilon^{2}y)f_{j}(x-2\varepsilon y)-f_{j}(x))\,dy,$

and Minkowski’s inequality gives

$\|h_{\varepsilon,j}\|_{L^{2}}\leq\int_{\mathbf{R}^{d}}|\psi(y)|\cdot\|(2\pi)^{\frac{d}{2}}\widehat{\psi}(\varepsilon\,\cdot\,-2\varepsilon^{2}y)f_{j}(\,\cdot\,-2\varepsilon
y)-f_{j}\|_{L^{2}}\,dy.$ (4.49)

We have $(2\pi)^{\frac{d}{2}}\widehat{\psi}(0)=1$ in view of (4.44), which
implies

$\lim_{\varepsilon\to
0+}\|(2\pi)^{\frac{d}{2}}\widehat{\psi}(\varepsilon\,\cdot\,-2\varepsilon^{2}y)f_{j}(\,\cdot\,-2\varepsilon
y)-f_{j}\|_{L^{2}}=0.$ (4.50)

In fact, the latter limit is evident when $f_{j}\in
C_{0}^{\infty}(\mathbf{R}^{d})$, and follows for $f_{j}\in
L^{2}(\mathbf{R}^{d})$ by density arguments, using that
$C_{0}^{\infty}(\mathbf{R}^{d})$ is dense in $L^{2}(\mathbf{R}^{d})$.

Furthermore, since

$(2\pi)^{\frac{d}{2}}\|\widehat{\psi}\|_{L^{\infty}}\leq\|\psi\|_{L^{1}}$

we obtain

$y\mapsto|\psi(y)|\cdot\|(2\pi)^{\frac{d}{2}}\widehat{\psi}(\varepsilon\,\cdot\,-2\varepsilon^{2}y)f_{j}(\,\cdot\,-2\varepsilon
y)-f_{j}\|_{L^{2}}\\\\[4.30554pt]
\leq|\psi(y)|(\|\psi\|_{L^{1}}+1)\|f_{j}\|_{L^{2}}\in L^{1}(\mathbf{R}^{d}).$
(4.51)

A combination of (4.49)–(4.51) and Lebesgue’s theorem now gives (4.48), and
the result follows. ∎

A combination of (4.10) and Theorem 4.13 gives the following. The details are
left for the reader.

######  Theorem 4.14.

Suppose $p\in(0,\sharp]$ and $A\in\mathbf{M}(d,\mathbf{R})$. Then

$\displaystyle
s_{p,A}(\mathbf{R}^{2d}){\text{\footnotesize$\\#$}}_{A}s_{\sharp,A}(\mathbf{R}^{2d})=s_{p,A}(\mathbf{R}^{2d}){\text{\footnotesize$\\#$}}_{A}s_{\infty,A}(\mathbf{R}^{2d})=s_{p,A}(\mathbf{R}^{2d}).$
(4.52)

######  Example 4.15.

Let $r\in(0,1]$. Due to (4.24), a natural question arise whether
$W\\!\\!L^{1,r}(\mathbf{R}^{2d})$ is contained in $s_{p}^{w}(\mathbf{R}^{2d})$
or not, for some choice of $p\in(0,\infty)$. Observe that for such $p$ we have
$s_{p}^{w}(\mathbf{R}^{2d})\subsetneq s_{\sharp}^{w}(\mathbf{R}^{2d})$. Our
results now give negative answer on this. That is, we have

$W\\!\\!L^{1,r}(\mathbf{R}^{2d})\nsubseteq\bigcup_{p\in\mathbf{R}_{+}}s_{p}^{w}(\mathbf{R}^{2d})\qquad\Big{(}\text{but}\quad
W\\!\\!L^{1,r}(\mathbf{R}^{2d})\subseteq
s_{\sharp}^{w}(\mathbf{R}^{2d})\Big{)}.$ (4.53)

In fact, let

$\mathscr{A}=s_{\sharp}^{w}(\mathbf{R}^{2d}),\quad\mathscr{A}_{0}=W\\!\\!L^{1,r}(\mathbf{R}^{2d})\quad\text{and}\quad\mathscr{A}_{t}=s_{1/t}^{w}(\mathbf{R}^{2d}),\
t>0,$

which are algebras under the twisted convolution. Now recall that
$\mathscr{A}_{0}=W\\!\\!L^{1,r}(\mathbf{R}^{2d})$ is a factorization algebra
under the twisted convolution and that Hölder’s inequality holds for the
$s_{p}^{w}$ classes in view of Theorem 2.5 and Proposition 4.10. Consequently,
the hypothesis of Proposition 1.11 is fulfilled for $G=(\mathbf{R}_{+},+)$
with the usual ordering. Since

$W\\!\\!L^{1,r}(\mathbf{R}^{2d})\nsubseteq
L^{2}(\mathbf{R}^{2d})=s_{2}^{w}(\mathbf{R}^{2d})=\mathscr{A}_{1/2},$

it now follows from Proposition 1.11 that (4.53) holds.

##  Appendix A Factorization properties for quasi-Banach modules

In this section we extend well-known factorization properties for Banach
algebras and modules to quasi-Banach algebras and modules. Especially we
follow the framework in [30, Section 2] and extend (2.5) Theorem in [30] in
such a way that we assume that the involved spaces are quasi-Banach algebras
and modules, instead of the more restrictive Banach algebras and modules.

In the first part we deduce some preparatory results which are needed for the
proof of the main result Theorem 1.5, which appear in the last part of the
section.

We have the following lemma, similar to (2.3) Lemma in [30]. Here in what
follows we let

$B_{r}=B_{r,\mathcal{B}}=B_{r}(0)=B_{r,\mathcal{B}}(0)\equiv\\{\,\phi\in\mathcal{B}\,;\,\|\phi\|_{\mathcal{B}}\leq
r\,\\}$

be the (closed) ball in $\mathcal{B}$, centered at $0$ and with radius $r$.

######  Lemma A.1.

Let $\mathcal{B}$ be a quasi-Banach algebra of order $p\in(0,1]$,
$\mathcal{M}$ be a left quasi-Banach $\mathcal{B}$ module of order $p$, let
$r\in[1,\infty)$ and let

$T:B_{r,\mathcal{B}}\to\mathcal{B}_{E}$

be given by

$\displaystyle T(\phi)$
$\displaystyle\equiv\left(\frac{2r+1}{2r}\right)\left(\varrho+\sum_{k=1}^{\infty}(-1)^{k}(2r)^{-k}\phi^{k}\right),$
$\displaystyle\quad\phi$ $\displaystyle\in B_{r,\mathcal{B}}.$ (A.1) Then
$\displaystyle T(\phi)$
$\displaystyle=\left(\frac{2r}{2r+1}\varrho+\frac{1}{2r+1}\phi\right)^{-1},$
$\displaystyle\quad\phi$ $\displaystyle\in B_{r,\mathcal{B}},$ (A.2)
$\displaystyle\|T(\phi)\cdot f-f\|_{\mathcal{M}}$
$\displaystyle\leq(1-2^{-p})^{-\frac{2}{p}}\,\frac{2r+1}{4r}\|\phi\cdot
f-f\|_{\mathcal{M}},$ $\displaystyle\quad\phi$ $\displaystyle\in
B_{r,\mathcal{B}},\ f\in\mathcal{M}$ (A.3) and
$\displaystyle\frac{1}{(2^{p}+1)^{\frac{1}{p}}}\Big{(}2\,+\,$
$\displaystyle\frac{1}{r}\Big{)}\leq\|T(\phi)\|_{\mathcal{B}_{E}}\leq\frac{1}{(2^{p}-1)^{\frac{1}{p}}}\Big{(}2+\frac{1}{r}\Big{)},$
$\displaystyle\quad\phi$ $\displaystyle\in B_{r,\mathcal{B}}.$ (A.4)

###### Proof.

The assertion (A.2) follows by straight-forward computations (see e. g. the
proof of (i) in (2.3) Lemma in [30]).

In order to prove (A.3) we observe that

$T(\phi)\cdot
f-f=\left(\frac{2r+1}{2r}\right)\sum_{k=1}^{\infty}(-1)^{k}(2r)^{-k}(\phi^{k}\cdot
f-f).$

By applying quasi-norms on the latter identity we obtain

$\|T(\phi)\cdot
f-f\|_{\mathcal{M}}^{p}\leq\left(\frac{2r+1}{2r}\right)^{p}\sum_{k=1}^{\infty}(2r)^{-kp}\|\phi^{k}\cdot
f-f\|_{\mathcal{M}}^{p}.$ (A.5)

For the norms in the sum we have

$\displaystyle\|\phi^{k}\cdot f-f\|_{\mathcal{M}}^{p}$
$\displaystyle\leq\sum_{j=1}^{k}\|\phi^{j}\cdot f-\phi^{j-1}\cdot
f\|_{\mathcal{M}}^{p}$
$\displaystyle\leq\sum_{j=1}^{k}\|\phi\|_{\mathcal{B}}^{(j-1)p}\|\phi\cdot
f-f\|_{\mathcal{M}}^{p}\leq kr^{kp}\|\phi\cdot f-f\|_{\mathcal{M}}^{p}.$

By inserting this into (A.5) we obtain

$\displaystyle\|T(\phi)\cdot f-f\|_{\mathcal{M}}^{p}$
$\displaystyle\leq\left(\frac{2r+1}{2r}\right)^{p}\left(\sum_{k=1}^{\infty}(2r)^{-kp}kr^{kp}\right)\|\phi\cdot
f-f\|_{\mathcal{M}}^{p}$
$\displaystyle=\left(\frac{2r+1}{2r}\right)^{p}\frac{2^{-p}}{(1-2^{-p})^{2}}\|\phi\cdot
f-f\|_{\mathcal{M}}^{p},$

which gives (A.3).

It remains to prove (A.4). Since

$\varrho=T(\phi)\left(\frac{2r}{2r+1}\varrho+\frac{1}{2r+1}\phi\right),$

we obtain

$\displaystyle 1$
$\displaystyle=\|\varrho\|_{\mathcal{B}_{E}}^{p}\leq\|T(\phi)\|_{\mathcal{B}_{E}}^{p}\left(\left(\frac{2r}{2r+1}\right)^{p}\|\varrho\|_{\mathcal{B}_{E}}^{p}+\left(\frac{1}{2r+1}\right)^{p}\|\phi\|_{\mathcal{B}_{E}}^{p}\right)$
$\displaystyle\leq\|T(\phi)\|_{\mathcal{B}_{E}}^{p}\left(\left(\frac{2r}{2r+1}\right)^{p}+\left(\frac{1}{2r+1}\right)^{p}r^{p}\right),$

which is the same as the left inequality in (A.4).

In similar ways we have

$\displaystyle\|T(\phi)\|_{\mathcal{B}_{E}}^{p}$
$\displaystyle=\left(\frac{2r+1}{2r}\right)^{p}\left(\|\varrho\|_{\mathcal{B}_{E}}^{p}+\sum_{k=1}^{\infty}(2r)^{-pk}\|\phi\|_{\mathcal{B}_{E}}^{pk}\right)$
$\displaystyle\leq\left(\frac{2r+1}{2r}\right)^{p}\left(1+\sum_{k=1}^{\infty}(2r)^{-pk}r^{pk}\right)=\left(\frac{2r+1}{2r}\right)^{p}\left(1+\frac{2^{-p}}{1-2^{-p}}\right),$

which is equivalent to the right inequality in (A.4), and the result follows.
∎

The next result corresponds to (2.4) Lemma in [30].

######  Lemma A.2.

Let $\mathcal{B}$ be a quasi-Banach algebra of order $p$, $\mathcal{M}$ be a
left quasi-Banach $\mathcal{B}$ module of order $p\in(0,1]$, $\varepsilon>0$
and $f\in\mathcal{M}$. Also assume that $\mathcal{B}$ possess a bounded left
approximate identity for $(\mathcal{B},\mathcal{M})$ of order
$r\in[1,\infty)$. Then there is a sequence
$\\{\phi_{n}\\}_{n=1}^{\infty}\subseteq B_{r,\mathcal{B}}$ such that if

$\displaystyle\psi_{n}$
$\displaystyle=\frac{(2r)^{n}}{(2r+1)^{n}}\varrho+\left(\sum_{k=1}^{n}\frac{(2r)^{k-1}}{(2r+1)^{k}}\phi_{k}\right)\in\mathcal{B}_{E},\quad
n\in\mathbf{N},$ (A.6)

then $\psi_{n}^{-1}$ exists in $\mathcal{B}_{E}$, and

$\|\psi_{n}^{-1}\cdot f-\psi_{n-1}^{-1}\cdot f\|_{\mathcal{M}}\leq
2^{-n}\varepsilon,\quad n\in\mathbf{Z}_{+}.$ (A.7)

Here we interpret the right-hand side of (A.6) as $\varrho$ when $n=0$. That
is,

$\psi_{0}=\varrho.$

###### Proof of Theorem 1.5.

By Definition 1.3 and (1.9), there is an element $\phi_{1}\in\mathcal{B}$ such
that ${\phi_{1}}\in B_{r,\mathcal{B}}$ and

$\|\phi_{1}\cdot
f-f\|_{\mathcal{M}}\leq(1-2^{-p})^{\frac{2}{p}}\,\frac{2r}{2r+1}\varepsilon.$

Then

$\psi_{1}=\frac{2r}{2r+1}\varrho+\frac{1}{2r+1}\phi_{1}.$

Hence Lemma A.1 shows that $\psi_{1}^{-1}=T(\phi_{1})$, and by (A.3) we get

$\|\psi_{1}^{-1}\cdot f-\psi_{0}^{-1}f\|_{\mathcal{M}}=\|T(\phi_{1})\cdot
f-f\|_{\mathcal{M}}\leq(1-2^{-p})^{-\frac{2}{p}}\,\frac{2r+1}{4r}\|\phi_{1}\cdot
f-f\|_{\mathcal{M}}\leq 2^{-1}\varepsilon,$

and the result follows for $n=1$.

Suppose the result holds true for $n\leq m$, $m\geq 1$. We shall prove the
result for $n=m+1$. Therefore, suppose that there are $\phi_{j}$,
$j=1,\dots,m$ such that $\psi_{j}^{-1}\in\mathcal{B}_{E}$ exists and satisfies
(A.7) for $j=1,\dots,m$.

Let $\phi\in B_{r,\mathcal{B}}$ be arbitrary and set

$\psi=\frac{(2r)^{m+1}}{(2r+1)^{m+1}}\varrho+\left(\sum_{k=1}^{m}\frac{(2r)^{k-1}}{(2r+1)^{k}}\phi_{k}\right)+\frac{(2r)^{m}}{(2r+1)^{m+1}}\phi,$

which we can write

$\psi=\left(\frac{2r}{2r+1}\varrho+\frac{1}{2r+1}\phi\right)\chi=T(\phi)^{-1}\chi,$
(A.8)

where

$\chi=\frac{(2r)^{m}}{(2r+1)^{m}}\varrho+\left(\sum_{k=1}^{m}\frac{(2r)^{k-1}}{(2r+1)^{k}}T(\phi)\phi_{k}\right).$

This gives

$\chi-\psi_{m}=\sum_{k=1}^{m}\frac{(2r)^{k-1}}{(2r+1)^{k}}(T(\phi)\phi_{k}-\phi_{k}).$
(A.9)

By Lemma A.1 with $\mathcal{M}$ replaced by $\mathcal{B}_{E}$ it follows that

$\displaystyle\|\chi-\psi_{m}\|_{\mathcal{B}_{E}}^{p}$
$\displaystyle\leq\sum_{k=1}^{m}\frac{(2r)^{p(k-1)}}{(2r+1)^{pk}}\|T(\phi)\phi_{k}-\phi_{k}\|_{\mathcal{B}_{E}}^{p}$
$\displaystyle\lesssim\sum_{k=1}^{m}\frac{(2r)^{p(k-1)}}{(2r+1)^{pk}}\|\phi\phi_{k}-\phi_{k}\|_{\mathcal{B}_{E}}^{p}$
(A.10)

Now let $\varepsilon_{1}>0$ be arbitrary. Then it follows from Definition 1.3,
Lemma A.1 and (A.10) that we may choose $\phi$ such that

$\displaystyle\|\chi-\psi_{m}\|_{\mathcal{B}_{E}}$
$\displaystyle<\varepsilon_{1},\quad\|\chi^{-1}-\psi_{m}^{-1}\|_{\mathcal{B}_{E}}<\varepsilon_{1}$
and $\displaystyle\|\phi\cdot f-f\|_{\mathcal{M}}$
$\displaystyle<\varepsilon_{1}.$

Here we have also used the fact that the invertibility of $\psi_{m}$ implies
that $\chi$ is invertible provided $\|\chi-\psi_{m}\|_{\mathcal{B}_{E}}$ is
small enough, and that the invertible elements in $\mathcal{B}_{E}$ is an open
set $\Omega\subseteq\mathcal{B}_{E}$, leading to that the map
$\chi\mapsto\chi^{-1}$ is a homeomorphism on $\Omega$. Here we have also used
the fact that we may choose $\phi\in B_{r,\mathcal{B}}$ such that

$\|\phi\phi_{k}-\phi_{k}\|_{\mathcal{B}_{E}}\quad\text{and}\quad\|\phi\cdot
f-f\|_{\mathcal{M}}$

can be made arbitrarily small.

This gives Hence, for such choice of $\phi$ it follows that $\psi$ in (A.8) is
invertible, and

$\displaystyle\|\psi^{-1}\cdot f-\psi_{m}^{-1}\cdot f\|_{\mathcal{M}}^{p}$
$\displaystyle=\|\chi^{-1}T(\phi)\cdot f-\psi_{m}^{-1}\cdot
f\|_{\mathcal{M}}^{p}$ $\displaystyle\leq\|\chi^{-1}T(\phi)\cdot
f-\psi_{m}^{-1}T(\phi)\cdot f\|_{\mathcal{M}}^{p}+\|\psi_{m}^{-1}T(\phi)\cdot
f-\psi_{m}^{-1}\cdot f\|_{\mathcal{M}}^{p}$
$\displaystyle\lesssim\|\chi^{-1}-\psi_{m}^{-1}\|_{\mathcal{B}_{E}}^{p}\|T(\phi)\|_{\mathcal{B}_{E}}^{p}\|f\|_{\mathcal{M}}^{p}+\|\psi_{m}^{-1}\|_{\mathcal{B}_{E}}^{p}\|T(\phi)\cdot
f-f\|_{\mathcal{M}}^{p}$
$\displaystyle\lesssim\|\chi^{-1}-\psi_{m}^{-1}\|_{\mathcal{B}_{E}}^{p}\|f\|_{\mathcal{M}}^{p}+\|\phi\cdot
f-f\|_{\mathcal{M}}^{p}<2^{-(m+1)}\varepsilon,$

provided $\varepsilon_{1}$ was chosen small enough. Hence, by choosing
$\phi_{m+1}=\phi$ and $\psi_{m+1}=\psi$, we have proved the result for
$n=m+1$, when it holds for $n\leq m$. The result now follows for general $n$
by induction. ∎

###### Proof.

Our strategy is to use the approximate identity to construct a sequence
$\\{\psi_{n}\\}_{n=1}^{\infty}$ of invertible elements in $\mathcal{B}_{E}$
such that the sequence $\\{\psi_{n}\\}_{n=1}^{\infty}$ converges to an element
in $\mathcal{B}$ and $\\{\psi_{n}^{-1}f\\}$ also converges in $\mathcal{M}$,
though $\\{\psi^{-1}_{n}\\}$ is an unbounded sequence in $\mathcal{B}$. The
elements $\psi$ and $g$ then appear from these sequences.

The result is obviously true for $f=0$. Therefore suppose that $f\neq 0$. Let
$k,n\in\mathbf{N}$, $\phi_{n}$ and $\psi_{n}$ be as in Lemma A.2, and set
$g_{n}=\psi_{n}^{-1}\cdot f$. Then $\psi_{n}\cdot g_{n}=f$, and (A.7) gives

$\displaystyle\|g_{n+k}-g_{n}\|_{\mathcal{M}}^{p}$
$\displaystyle\leq\sum_{j=0}^{k-1}\|g_{n+j+1}-g_{n+j}\|_{\mathcal{M}}^{p}$
(A.11)
$\displaystyle<\varepsilon^{p}\sum_{j=0}^{k-1}2^{-p(n+j)}<\varepsilon^{p}\frac{2^{-pn}}{1-2^{-p}}\to
0,$

as $k,n\to\infty$. Hence $\\{g_{n}\\}_{n=1}^{\infty}$ is a Cauchy sequence
which has a limit $g\in\mathcal{M}$.

By (1.9) it follows that $f\in\mathcal{M}_{f}$. Hence
$g_{n}=\psi_{n}^{-1}\cdot f\in\mathcal{M}_{f}$ and $g\in\mathcal{M}_{f}$,
since $\mathcal{B}_{E}\cdot\mathcal{M}_{f}\subseteq\mathcal{M}_{f}$ and that
$\mathcal{M}_{f}$ is closed.

By letting $n=0$ and $k$ tending to $\infty$, (A.11) gives

$\|g-f\|_{\mathcal{M}}\leq\frac{\varepsilon}{(1-2^{-p})^{\frac{1}{p}}},$

and (2) follows after redefining $\varepsilon>0$. It follows from (A.6) that

$\psi\equiv\lim_{n\to\infty}\psi_{n}=\sum_{k=1}^{\infty}\frac{(2r)^{k-1}}{(2r+1)^{k}}\phi_{k},$
(A.12)

where the limit exists in $\mathcal{B}_{E}$. Since the series on the right-
hand side in (A.12) converges in $\mathcal{B}$, we have $\psi\in\mathcal{B}$.
Furthermore,