# Spectral Invariance of Quasi-Banach Algebras of Matrices and
Pseudodifferential Operators

Karlheinz Gröchenig  Faculty of Mathematics
University of Vienna
Oskar-Morgenstern-Platz 1
A-1090 Vienna, Austria<EMAIL_ADDRESS>,  Christine
Pfeuffer  Faculty of Mathematics
Martin-Luther-Universität Halle-Wittenberg
06099 Halle (Saale), Germany<EMAIL_ADDRESS>and
Joachim Toft  Faculty of Mathematics
Lineaus University
Universitetsplatsen 1
35195 Växjö, Sweden<EMAIL_ADDRESS>
###### Abstract.

We extend the stability and spectral invariance of convolution-dominated
matrices to the case of quasi-Banach algebras $p<1$. As an application we
construct a spectrally invariant quasi-Banach algebra of pseudodifferential
operators with non-smooth symbols that generalize Sjöstrand’s results.

###### Key words and phrases:

pseudodifferential operators, spectral invariance, modulation space, Wiener’s
lemma, off-diagonal decay matrices, Gabor frame

###### 2010 Mathematics Subject Classification:

35P05, 35S05, 47G30

##  1\. Introduction

Spectral invariance is an important phenomenon for applications in the field
of partial differential equations and in the theory of pseudodifferential
operators. The first result due to Beals [6] asserts that the inverse of a
pseudodifferential operator $T$ that is (i) invertible on
$L^{2}(\mathbb{R}^{d})$ and (ii) with a symbol in the Hörmander class
$S^{0}_{0,0}$ is again a pseudodifferential operator with a symbol in the same
class. In other words, this class of pseudodifferential operators is inverse-
closed (closed under inversion) in the algebra $\mathcal{A}$ of
pseudodifferential operators with $S^{0}_{0,0}$-symbols. As a consequence the
spectrum of $T$ is independent of the weighted $L^{p}(\mathbb{R}^{d})$ space
or of the choice of $B^{s,a}_{p,q}(\mathbb{R}^{d})$, see [32, 36]. This
phenomenon is often referred to as spectral invariance, and the resemblance to
Wiener’s lemma for absolutely convergent Fourier series has also motivated the
terminology of $\mathcal{A}$ being a Wiener algebra.

The next important step in the theory of spectral invariance of
pseudodifferential operators was made by Sjöstrand [38] who introduced a class
of non-smooth symbols for which the associated algebra of pseudodifferential
operators is spectrally invariant in $\mathcal{B}(L^{2}(\mathbb{R}^{d}))$.
This class, nowadays called the Sjöstrand class, turned out to be an already
known function space that is paramount in time-frequency analysis, namely the
modulation space $M^{\infty,1}({\mathbb{R}^{2d}})$. This connection spawned an
intensive investigation of pseudodifferential operators with time-frequency
methods [19, 24, 26, 21, 14, 40, 39, 42, 43, 38, 35]. Among the new results
obtained by time-frequency methods was, firstly, a characterization of the
Hörmander class and the Sjöstrand class by means of the matrix associated to a
pseudodifferential operator with respect to a Gabor frame. Secondly, time-
frequency analysis established a firm connection between the off-diagonal
decay of these matrices and the corresponding properties (boundedness,
algebra, inverse-closedness) of pseudodifferential operators. In this way,
every (solid) inverse-closed subalgebra of
$\mathcal{B}(\ell^{2}({\mathbb{Z}^{2d}}))$ can be mapped to an algebra of
pseudodifferential operators that is inverse-closed in
$\mathcal{B}(L^{2}(\mathbb{R}^{d}))$ [24]. In contrast to the classical hard
analysis methods the time-frequency approach is so flexible that the theory
can even be formulated for pseudodifferential operators on locally compact
abelian groups [26].

The goal of this paper is the extension of the theory of spectrally invariant
algebras of pseudodifferential operators to the realm of quasi-Banach
algebras. Quasi-Banach algebras are interesting in their own right, but quasi-
Banach spaces and associated operators occur naturally in approximation theory
and data compression problems, see e.g. [11, 25]. Additionally in time-
frequency analysis they occur in the formulation of uncertainty principles
[16].

To formulate our main results, we briefly recall the definition of modulation
spaces and the Weyl form of pseudodifferential operators. For a fixed non-zero
Schwartz function $g\in\mathcal{S}({\mathbb{R}^{d}})$ and a tempered
distribution $f$ the short-time Fourier transform $V_{g}f$ is the function on
$\mathbb{R}^{2d}$ defined by the formula

$\displaystyle
V_{g}f(x,\xi):=\int_{\mathbb{R}^{d}}f(t)\overline{g(t-x)}e^{-2\pi
it\cdot\xi}dt\,$ (1.1)

(with suitable interpretation of the integral). For $0<p,q\leq\infty$ the
(unweighted) _modulation space_ $M^{p,q}(\mathbb{R}^{d})$ is defined by the
quasi-norm

$\displaystyle\|f\|_{M^{p,q}}:=\left(\int_{\mathbb{R}^{d}}\left(\int_{\mathbb{R}^{d}}|V_{g}f(x,\xi)|^{p}\,dx\right)^{q/p}d\xi\right)^{1/q}$
(1.2)

(with usual modifications in case $p=\infty$ or $q=\infty$) and consists of
all tempered distributions $f$ with finite quasi-norm. Modulation spaces on
${\mathbb{R}^{2d}}$ serve as symbol classes for pseudodifferential operators.
Our focus will be on the symbol class $M^{\infty,p_{0}}({\mathbb{R}^{2d}})$
for $p_{0}<1$. This is only a quasi-Banach space.

Given a symbol $\mathfrak{a}$ on ${\mathbb{R}^{2d}}$, the corresponding
pseudodifferential operator in the Weyl calculus is defined formally as

$\mathfrak{a}^{w}f(x)=\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\mathfrak{a}\big{(}\frac{x+y}{2},\xi)f(y)e^{2\pi
i(x-y)\cdot\xi}\,dyd\xi\,,$

(again with a suitable interpretation of the integral and for
$f\in\mathcal{S}(\mathbb{R}^{d})$).

With these definitions our main results can be stated as follows.

######  Theorem 1.1 (Spectral invariance).

Let $p_{0}\in(0,1]$ and $\mathfrak{a}\in M^{\infty,p_{0}}({\mathbb{R}^{2d}})$
be such that $\mathfrak{a}^{w}$ is invertible on $M^{p}(\mathbb{R}^{d})$ for
some $p\in[p_{0},\infty]$. Then $(\mathfrak{a}^{w})^{-1}=\mathfrak{b}^{w}$ for
some $\mathfrak{b}\in M^{\infty,p_{0}}({\mathbb{R}^{2d}})$.

As a consequence we obtain that the spectrum of a pseudodifferential operator
with a symbol in $M^{\infty,p_{0}}({\mathbb{R}^{2d}})$ is independent of the
space on which it acts.

######  Theorem 1.2 (Spectral invariance on Modulation Spaces).

If $\mathfrak{a}\in M^{\infty,p_{0}}({\mathbb{R}^{2d}})$ for $p_{0}\in(0,1]$
and $\mathfrak{a}^{w}$ is invertible on $M^{p}(\mathbb{R}^{d})$ for some
$p\in[p_{0},\infty]$, then $\mathfrak{a}^{w}$ is also invertible on
$M^{q}(\mathbb{R}^{d})$ for all $q\in[p_{0},\infty)$.

More generally, we show in Theorem 4.10 the invertibility of
$\mathfrak{a}^{w}$ on the more broad class of modulation spaces
$M^{r,q}(\mathbb{R}^{d})$ with $r,q\in[p_{0},\infty)$ under the assumptions of
the previous theorem.

Methods. We follow the proof outline of [24]. The first step is to study the
matrix representation of a pseudodifferential operator with respect to a Gabor
frame and then derive a characterization of the symbol class in terms of the
off-diagonal decay of the associated matrix.

In the second step this leads to the study of spectrally invariant matrix
algebras. The appropriate class in our context are convolution-dominated
matrices, i.e., matrices $A=(a_{\lambda,\rho})_{\lambda,\rho\in\Lambda}$ with
an off-diagonal decay of the form

$|a_{\lambda,\rho}|\leq H(\lambda-\rho)$

for a (smooth) function $H$ in $L^{p}(\mathbb{R}^{d})$. It turns out that such
matrix classes are spectrally invariant in $\mathcal{B}(\ell^{p}(\Lambda))$.
To offer a glimpse of this aspect, we formulate a very special case of our
main result on matrices that does not require technical details.

######  Theorem 1.3.

Let $A=(a_{kl})_{k,l\in\mathbb{Z}^{d}}$ be a matrix over the index set
$\mathbb{Z}^{d}$. Assume that there exists a sequence
$h\in\ell^{p_{0}}(\mathbb{Z}^{d})$ for $0<p_{0}\leq 1$, such that

$|a_{kl}|\leq h(k-l)\qquad\text{ for all }k,l\in\mathbb{Z}^{d}\,.$

(i) Spectral invariance: If $A$ is invertible on some
$\ell^{p}(\mathbb{Z}^{d})$ for $p\in[p_{0},\infty]$, then $A$ is invertible on
all $\ell^{q}(\mathbb{Z}^{d})$ for $q\in[p_{0},\infty]$.

(ii) Spectral Stability: If for some $\ell^{p}(\mathbb{Z}^{d})$ with
$p\in[p_{0},\infty]$, $A$ satisfies the stability condition

$\|Ac\|_{p}\geq C\|c\|_{p}\text{ for all }c\in\ell^{p}(\mathbb{Z}^{d})\,,$

then $A$ satisfies $\|Ac\|_{q}\geq C_{q}\|c\|_{q}\text{ for all
}c\in\ell^{q}(\mathbb{Z}^{d})$ with $q\in[p_{0},\infty]$.

In our approach we follow Sjöstrand ingenious proof of Wiener’s lemma for
absolutely convergent Fourier series [38] and built on the presentation in
[22]. Our ultimate results on spectral invariance and stability (Theorems 3.5
and 3.14) are a significant extension of the above preliminary statement and
provide several new facts of spectral invariance:

  * (i)

they yield both spectral stability and spectral invariance,

  * (ii)

they are formulated with respect to arbitrary operator algebras
$\mathcal{B}(\ell^{p})$ (not just $\mathcal{B}(\ell^{2})$ as is usually done)
and

  * (iii)

they cover the general case of quasi-Banach algebras, and

  * (iv)

in addition we treat arbitrary index sets and not just $\mathbb{Z}^{d}$ or a
discrete abelian group as in most references.

Technically, the study of quasi-Banach algebras of convolution-dominated
matrices is the main part of our paper, its application to pseudodifferential
operators is then based on our analysis in [24]. Our proof contains some new
features and avoids the functional calculus associated with the pseudo-
inverse. These arguments may be useful in other contexts as well.

Related results. There are numerous results on the spectral invariance of
matrices, we mention [1, 3, 4, 41, 40, 39, 37, 38, 30] for a small sample and
[20] for a survey. As long as the index set is a discrete abelian group, one
can use methods from harmonic analysis to establish spectral invariance. This
line of thought goes back to Bochner and Philipps and is used in [3, 4, 10, 8]
and many others. All these proofs break down, however, when unstructured index
sets are considered.

The extension of spectral invariance to quasi-Banach algebras of matrices and
operators over $\mathbb{Z}^{d}$ is the subject of the recent papers [8, 10].
While there is some thematic overlap, not all results are directly comparable.
On the one hand, we restrict our attention to unweighted $\ell^{p}$-spaces,
whereas [8, 10] include weights. On the other hand, these papers prove that
invertibility of convolution-dominated matrices on the Hilbert space
$\ell^{2}$ implies invertibility on all $\ell^{p}$ for $p>p_{0}$. Our results
also provide a converse, namely that invertilibity on some $\ell^{p}$ implies
invertibility on $\ell^{2}$.

As for the application to pseudodifferential operators, the Banach algebra
case $p_{0}=1$ for the symbols with invertibility on
$M^{2,2}=L^{2}(\mathbb{R}^{d})$ was already proved by Sjöstrand [38]. The
quasi-Banach algebra case $p_{0}<1$ with invertibility on
$M^{2,2}=L^{2}(\mathbb{R}^{d})$ was established in [8] based on the time-
frequency methods developed in [24]. We want to emphasize that the general
case where we start with invertibility on $M^{p}(\mathbb{R}^{d}),p\neq 2$, is
much more involved, as there are no Hilbert space techniques available. For
self-adjoint pseudodifferential operators
$(\mathfrak{a}^{w})^{*}=\mathfrak{a}^{w}$ one could argue with duality and
interpolation to reduce to the case of invertibility on
$L^{2}(\mathbb{R}^{d})$, but there is no cheap trick for non-self-adjoint
pseudodifferential operators. When we start with invertibility on a quasi-
Banach space $M^{p}(\mathbb{R}^{d}),p<1$, even duality is no longer useful.
For this reason we disregarded the methods of Bochner-Philipps as used in [10,
24, 8] for the construction of spectrally invariant matrix algebras and
instead exploit Sjöstrand’s original proof of Wiener’s lemma in [38].

The paper is organized as follows: In Section 2 we collect the relevant
definitions about sequence spaces and amalgam spaces. In Section 3 we treat
the spectral invariance of convolution-dominated matrices. We treat both the
stability of such matrices on the quasi-Banach spaces $\ell^{p},p<1$ and the
spectral invariance of the algebra of convolution-dominated matrices. The main
results are Theorems 3.5 and 3.14. In Section 4 we first recapitulate the
definitions of modulation spaces and Gabor frames and various calculi of
pseudodifferential operators and then prove our main theorems that are already
stated in the introduction. For completeness we have postponed some easy and
known proofs to the appendix.

##  2\. Preliminaries

For the convenience of the reader we now list up the definitions of some
function spaces and their properties needed during this paper. We start with
the sequence spaces.

###  2.1. Sequence space $\ell^{p}$

For each $0<p\leq\infty$ and discrete set $J$, we recall that the set
$\ell^{p}(J)$ consists of all complex-valued sequences $a$ such that

$\displaystyle\|a\|_{p}:=\left(\sum_{j\in J}|a_{j}|^{p}\right)^{1/p},\qquad
a=(a_{j})_{j\in J}$

is finite (with usual modifications for $p=\infty$). Then $\ell^{p}(J)$ is a
quasi-Banach space with quasi-norm $\|\,\cdot\,\|_{p}$, which is even a norm
if $p\geq 1$.

We recall the following properties for $\ell^{p}(J)$.

######  Lemma 2.1.

Suppose $0<p\leq 1$, $J$ is a discrete set, $\Lambda$ is countable. Let
$a=(a_{j})_{j\in J}\in\ell^{p}(J)$ and $b,b_{\lambda}\in\ell^{p}(J)$,
$\lambda\in\Lambda$. Then the following is true:

  1. (i)

$\|a\|_{1}^{p}\leq\|a\|_{p}^{p}$, or $|\sum_{j\in J}a_{j}|^{p}\leq\sum_{j\in
J}|a_{j}|^{p}$;

  2. (ii)

if $\sum_{\lambda\in\Lambda}\|b_{\lambda}\|^{p}_{p}<\infty$, then
$\sum_{\lambda\in\Lambda}b_{\lambda}$ is uniquely defined as an element in
$\ell^{p}(J)$, and
$\|\sum_{\lambda\in\Lambda}b_{\lambda}\|_{p}^{p}\leq\sum_{\lambda\in\Lambda}\|b_{\lambda}\|_{p}^{p}$;

  3. (iii)

If $J=\mathbb{Z}^{d}$, then also $\|a\ast b\|_{p}\leq\|a\|_{p}\,\|b\|_{p}$.

###  2.2. Wiener Amalgam Space

Let $X=L^{\infty}(\mathbb{R}^{d})$ or
$X=C_{b}(\mathbb{R}^{d}):=L^{\infty}(\mathbb{R}^{d})\cap C(\mathbb{R}^{d})$,
and let $K\subseteq\mathbb{R}^{d}$ be convex and compact with positive volume.
The Wiener Amalgam space $W(X,L^{p_{0}})$ for $0<p_{0}\leq\infty$ consists of
all $f\in X$ such that

$\displaystyle\bigl{\lVert}f\bigr{\rVert}_{K,W(X,L^{p_{0}})}:=\left(\int_{\mathbb{R}^{d}}\|f\|^{p_{0}}_{L^{\infty}(x+K)}\,dx\right)^{1/p_{0}}<\infty\,.$

This is always a quasi-norm, and a norm, if $p_{0}\geq 1$ (see e.g. [28]). For
$p_{0}=\infty$ we have $W(X,L^{\infty})=X$. By compactness it follows that
$W(X,L^{p_{0}})$ is independent of the choice of $K$, and different $K$ yield
equivalent quasi-norms. For convenience we set
$\bigl{\lVert}\,\cdot\,\bigr{\rVert}_{W(X,L^{p_{0}})}=\bigl{\lVert}\,\cdot\,\bigr{\rVert}_{B_{1}(0),W(X,L^{p_{0}})}$.

######  Remark 2.1.

We observe that every continuous function with compact support is contained in
$W(C_{b},L^{p})$ for all $p>0$ and that $W(C_{b},L^{p})$ is translation
invariant (see e. g. [7] and the references therein).

The Wiener Amalgam space $W(C_{b},L^{p_{0}})$, $0<p_{0}<\infty$ arises
naturally in the formulation of sampling inequalities. We first recall that a
set $\Lambda\subseteq\mathbb{R}^{d}$ is _relatively separated_ , if

$\displaystyle\mathop{\mathrm{rel}}(\Lambda):=\sup\\{\\#(\Lambda\cap
B_{1}(x)):x\in\mathbb{R}^{d}\\}<\infty.$ (2.1)

######  Lemma 2.2.

Let $0<p_{0}<\infty$, $\Lambda\subseteq\mathbb{R}^{d}$ be relatively separated
and let $H\in W(C_{b},L^{p_{0}})$. Then
$(H(\lambda))_{\lambda\in\Lambda}\in\ell^{p_{0}}(\Lambda)$.

Lemma 2.2 follows by straight-forward estimates. In order to be self-
contained, a proof of the result is given in Appendix A.

##  3\. Spectral Invariance of Convolution-Dominated Matrices

In this section we prove a spectral invariance result for infinite dimensional
convolution-dominated matrices. For this we first list some needed auxiliary
tools.

We always denote the conjugate exponent of $p\in[1,\infty]$ by
$p^{\prime}=\tfrac{p}{p-1}$, so that $p^{\prime}\in[1,\infty]$ and

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

We identify a matrix $A=(a_{\lambda,\rho})_{\lambda\in\Lambda,\rho\in\Pi}$
indexed by $\Lambda$ and $\Pi$ with a linear operator

$\displaystyle(Ab)_{\lambda}:=\sum_{\rho\in\Pi}a_{\lambda\rho}b_{\rho}\qquad\text{
for }b=(b_{\rho})_{\rho\in\Pi}\,.$

Then $A$ is always well-defined on finite sequences and $A$ maps finite
sequences on $\Pi$ to arbitrary sequences on $\Lambda$. Some boundedness
properties of matrices on $\ell^{p}$-spaces are given by Schur’s test.

######  Proposition 3.1 (Schur test for $p\geq 1$).

Let $1\leq p\leq\infty$, $\Lambda$ and $\Pi$ be countable sets,
$A=(a_{\lambda,\rho})_{\lambda\in\Lambda,\rho\in\Pi}\in\mathbb{C}^{\Lambda\times\Pi}$
be a matrix satisfying

$\sup_{\rho\in\Pi}\sum_{\lambda\in\Lambda}|a_{\lambda,\rho}|\leq
K_{1}\quad\text{and}\quad\sup_{\lambda\in\Lambda}\sum_{\rho\in\Pi}|a_{\lambda,\rho}|\leq
K_{2}\,.$ (3.1)

Then $A$ is a bounded operator from $\ell^{p}(\Pi)$ to $\ell^{p}(\Lambda)$,
and

$\lVert A\rVert_{\mathcal{B}(\ell^{p}(\Pi),\ell^{p}(\Lambda))}\leq
K_{1}^{1/p^{\prime}}K_{2}^{1/p}.$

For a proof of Proposition 3.1, see e. g. [18, Lemma 6.1.2].

Because of (3.1) we let

$\|A\|_{\operatorname{Schur}}=\sup_{\rho\in\Pi}\sum_{\lambda\in\Lambda}|a_{\lambda,\rho}|+\sup_{\lambda\in\Lambda}\sum_{\rho\in\Pi}|a_{\lambda,\rho}|,$
(3.2)

for every $A\in\mathbb{C}^{\Lambda\times\Pi}$.

The following quasi-Banach space version of the Schur test just follows by
using the triangle inequality.

######  Proposition 3.2 (Schur test for $p\leq 1$).

Fix $0<p\leq 1$. Let
$A=(a_{\lambda,\rho})_{\lambda\in\Lambda,\rho\in\Pi}\in\mathbb{C}^{\Lambda\times\Pi}$
be a matrix satisfying

$\|A\|_{S-p}:=\sup_{\rho\in\Pi}\sum_{\lambda\in\Lambda}|a_{\lambda,\rho}|^{p}<\infty\,.$
(3.3)

Then $A$ defines a bounded operator from $\ell^{p}(\Pi)$ to
$\ell^{p}(\Lambda)$. The operator norm is bounded by $\|A\|_{S-p}^{1/p}$. If
$0<q\leq p\leq 1$, then

$\|A\|_{S-p}^{q/p}=\Big{(}\sup_{\rho\in\Pi}\sum_{\lambda\in\Lambda}|a_{\lambda,\rho}|^{p}\Big{)}^{q/p}\leq\sup_{\rho\in\Pi}\sum_{\lambda\in\Lambda}|a_{\lambda,\rho}|^{q}=\|A\|_{S-q}\,.$
(3.4)

Our treatment of the spectral invariance of pseudodifferential operators
relies on spectral invariance properties of associated convolution-dominated
matrices. Roughly speaking, a matrix $A\in\mathcal{B}(\ell^{2}(\Lambda))$ is
called _convolution-dominated_ , if there exists a function $H$, such that

$|a_{\lambda,\rho}|\leq
H(\lambda-\rho)\qquad\forall\lambda\in\Lambda,\rho\in\Pi\,.$ (3.5)

The function $H$ is then called an _envelope_ of $A$. By specifying a norm on
envelopes, we define a particular class of convolution-dominated matrices as
follows.

######  Definition 3.3.

Let $\Lambda,\Pi\subseteq\mathbb{R}^{d}$ be relatively separated and
$0<p_{0}\leq 1$. The set
$\mathcal{C}^{p_{0}}=\mathcal{C}^{p_{0}}(\Lambda,\Pi)$ consists of all
convolution-dominated matrices $A$ such that (3.5) holds for an envelope $H\in
W(C_{b},L^{p_{0}})$. For $A=(a_{\lambda,\rho})_{\lambda\in\Lambda,\rho\in\Pi}$
we set

$\displaystyle\|A\|_{\mathcal{C}^{p_{0}}}=\inf\\{\|H\|_{W(C_{b},L^{p_{0}})}:|a_{\lambda,\rho}|\leq
H(\lambda-\rho),\forall\lambda\in\Lambda,\rho\in\Pi\\}.$

We set $\mathcal{C}^{p_{0}}(\Lambda)=\mathcal{C}^{p_{0}}(\Lambda,\Lambda)$. We
observe that if $\Lambda=\Pi\subseteq\mathbb{R}^{d}$ is a lattice, then the
restriction of $H\in W(C_{b},L^{p_{0}})$ to $\Lambda$ belongs to
$\ell^{p_{0}}(\Lambda)$ due to Lemma 2.2. In this case,
$A\in\mathcal{C}^{p_{0}}(\Lambda)$, if and only if there is a sequence
$H\in\ell^{p_{0}}(\Lambda)$, such that $|A_{\lambda,\rho}|\leq
H(\lambda-\rho)$ for all $\lambda,\rho\in\Lambda$.

We note that $\|A\|_{\mathcal{C}^{p_{0}}}$ is a quasi-norm for $p_{0}<1$, and
$\mathcal{C}^{p_{0}}$ is a quasi-Banach $*$-algebra (sometimes called a
$p$-algebra) with respect to addition and multiplication of matrices. For
$p_{0}=1$, $\mathcal{C}^{1}$ is a Banach $*$-algebra with norm
$\|\cdot\|_{\mathcal{C}^{1}}$.

For convolution-dominated operators in Definition 3.3 we state the following
boundedness result. Here we set $q^{\prime}=\infty$ when $q\leq 1$

######  Proposition 3.4.

Let $0<p_{0}\leq 1$, $\Lambda$ and $\Pi$ be as in Definition 3.3 and let
$A\in\mathcal{C}^{p_{0}}(\Lambda,\Pi)$. Then $A$ is bounded from
$\ell^{q}(\Pi)$ to $\ell^{q}(\Lambda)$ for every $q\in[p_{0},\infty]$, and

$\displaystyle\|A\|_{\mathcal{B}(\ell^{q}(\Pi),\ell^{q}(\Lambda))}\leq
C\operatorname{rel}(\Lambda)^{\frac{1}{q}}\operatorname{rel}(\Pi)^{\frac{1}{q^{\prime}}}\|A\|_{\mathcal{C}^{p_{0}}(\Lambda,\Pi)},$
(3.6)

where the constant $C>0$ only depends on $d$.

The result follows by suitable combinations of Hölder’s and Young’s
inequalities. In order to be self-contained we present a proof in Appendix A.

###  3.1. Invariance of the lower bound property on $\ell^{p}$ of convolution-
dominated matrices

Our main technical contribution is the so-called stability of convolution-
dominated matrices. By this is meant the invariance of the lower bound
property of such matrices on $\ell^{p}$.

######  Theorem 3.5.

Let $p_{0}\leq 1$, $q\in[p_{0},\infty]$, $\Lambda,\Pi\subseteq\mathbb{R}^{d}$
be relatively separated, and $A\in\mathcal{C}^{p_{0}}(\Lambda,\Pi)$. Assume
that there exists $p\in[p_{0},\infty]$ and $C_{0}>0$ such that

$\lVert Ac\rVert_{p}\geq C_{0}\lVert c\rVert_{p},\qquad c\in\ell^{p}(\Pi)\,.$
(3.7)

Then there exists a constant $C>0$ which is independent of $q$ such that

$\lVert Ac\rVert_{q}\geq C\lVert c\rVert_{q}\qquad\text{ for all
}c\in\ell^{q}(\Pi)\,.$ (3.8)

In other words, if $A$ in Theorem 3.5 is bounded from below on some $\ell^{p}$
with $p\geq p_{0}$, then $A$ is bounded from below on $\ell^{q}$ for all
$q\in[p_{0},\infty]$. Note that (3.7) is equivalent to saying that $A$ is one-
to-one on $\ell^{p}(\Lambda)$ with closed range in $\ell^{p}(\Pi)$. Thus if
$A$ is one-to-one with closed range for _some_ $p\in[p_{0},\infty]$, then it
is one-to-one with closed range for _all_ $p\in[p_{0},\infty]$.

Note that by (3.6) $A$ in Theorem 3.5 is bounded from $\ell^{p}(\Pi)$ to
$\ell^{p}(\Lambda)$ for all $p\geq p_{0}$, hence the estimates (3.7) and (3.8)
really make sense.

The proof of Theorem 3.5 is modelled on Sjöstrand’s treatment of Wiener’s
lemma for convolution-dominated matrices. It exploits the flexibility of
Sjöstrand’s methods to transfer lower bounds for a matrix from one value of
$p$ to all others.

######  Remark 3.1.

The proof of the previous Proposition for $p_{0}=1$ can be found in [22,
Proposition 8.1]. Hence we can restrict ourselves to the case $p_{0}<1$. The
cases to be considered are:

  * (i)

from $p\geq 1$ to $q\geq 1$;

  * (ii)

from $p\leq 1$ to $q<p$;

  * (iii)

from $p\geq 1$ to $q<1$;

  * (iv)

from $p<1$ to $p<q$.

Case (iii) is a consequence of cases (i) and (ii). If $p\geq 1$ we can assume
$p=1$ on account of case (i). For $p=1$ the statement of case (iii) is
included in case (ii).

We need some preparations for the proof. First let $\varphi\in
C^{\infty}(\mathbb{R}^{d})$ with $0\leq\varphi\leq 1$ and
$\operatorname{supp}(\varphi)\subseteq B_{2}(0)$ be such that
$\\{\varphi(\cdot-k)\\}_{k\in\mathbb{Z}^{d}}$ is a partition of unity. Set
$\varphi^{\varepsilon}_{k}(x):=\varphi(\varepsilon x-k),\
x,y\in\mathbb{R}^{d}$. Then

$\displaystyle\sum_{k\in\mathbb{Z}^{d}}\varphi^{\varepsilon}_{k}$
$\displaystyle=1,\quad\lvert\varphi^{\varepsilon}_{k}(x)-\varphi^{\varepsilon}_{k}(y)\rvert\lesssim\varepsilon\lvert
x-y\rvert,\quad 0\leq\varphi^{\varepsilon}_{k}\leq 1\quad\text{and}$ (3.9)
$\displaystyle\Phi^{\varepsilon}$
$\displaystyle:=\sum_{k\in\mathbb{Z}^{d}}\left(\varphi^{\varepsilon}_{k}\right)^{2}\asymp
1\,.$

By combining these properties, we obtain

$\displaystyle\lvert\varphi^{\varepsilon}_{k}(x)-\varphi^{\varepsilon}_{k}(y)\rvert\lesssim\min\\{1,\varepsilon|x-y|\\},\qquad
x,y\in\mathbb{R}^{d}.$ (3.10)

For $k\in\mathbb{Z}^{d}$ and $\varepsilon>0$ let
$\varphi^{\varepsilon}_{k}c:={\varphi^{\varepsilon}_{k}}\big{|}_{\Pi}\cdot
c,c=(c_{\rho})_{\rho\in\Pi}$, denote the multiplication operator
$\varphi^{\varepsilon}_{k}$. This multiplication operator enables us to get
equivalent norms for sequence spaces.

######  Lemma 3.6.

Let $\varepsilon>0$ and $\Lambda\subseteq\mathbb{R}^{2d}$ be relatively
separated. Then for $0<q\leq\infty$ we get

$\displaystyle\Big{(}\sum_{k\in\mathbb{Z}^{d}}\lVert\varphi^{\varepsilon}_{k}a\rVert_{q}^{q}\Big{)}^{1/q}\asymp\lVert
a\rVert_{q},\qquad a\in\ell^{q}(\Pi),$ (3.11)

with the usual modifications in case $q=\infty$.

For $1\leq q\leq\infty$, Lemma 3.6 was already proved in [22], and the other
cases are obtained by similar arguments. For completeness we present a proof
for $q<1$ in Appendix A.

######  Lemma 3.7.

Let $0<p_{0}\leq 1$, $p,q\in[p_{0},\infty]$, $\varepsilon>0$ and
$\Lambda\subseteq\mathbb{R}^{2d}$ be relatively separated. Then

$\displaystyle\sum_{k\in\mathbb{Z}^{d}}\lVert\varphi^{\varepsilon}_{k}a\rVert_{p}^{q}\asymp\lVert
a\rVert_{q}^{q},\qquad a\in\ell^{q}(\Pi),$ (3.12)

with the usual modifications in case $p=\infty$ or $q=\infty$. The constants
in (3.12) are independent of $p,q$, but depend on $\varepsilon$.

In the case $p,q\geq 1$ the claim was already shown in [22].

###### Proof.

Let $p,q\in[p_{0},\infty]$ be arbitrary. For fixed $\varepsilon>0$ we get

$\displaystyle
N:=\sup_{k\in\mathbb{Z}^{d}}\\#\operatorname{supp}({\varphi^{\varepsilon}_{k}}\big{|}_{\Pi})=\sup_{k\in\mathbb{Z}^{d}}\\#\big{\\{}\,\rho\in\Pi\,:\,\varphi^{\varepsilon}_{k}(\rho)\not=0\,\big{\\}}<\infty.$

Since $q\geq p_{0}$ we have

$\displaystyle\lVert\varphi^{\varepsilon}_{k}a\rVert_{q}\leq\lVert\varphi^{\varepsilon}_{k}a\rVert_{p_{0}}\leq
N^{1/p_{0}}\lVert\varphi^{\varepsilon}_{k}a\rVert_{\infty}\leq
N^{1/p_{0}}\lVert\varphi^{\varepsilon}_{k}a\rVert_{p},\qquad
a\in\ell^{\infty}(\Pi),$

and similarly

$\displaystyle\lVert\varphi^{\varepsilon}_{k}a\rVert_{p}\leq
N^{1/p_{0}}\lVert\varphi^{\varepsilon}_{k}a\rVert_{q},\qquad
a\in\ell^{\infty}(\Pi).$

As a consequence we obtain for $q\neq\infty$,

$\displaystyle\Big{(}\sum_{k\in\mathbb{Z}^{d}}\lVert\varphi^{\varepsilon}_{k}a\rVert_{p}^{q}\Big{)}^{1/q}\asymp\Big{(}\sum_{k\in\mathbb{Z}^{d}}\lVert\varphi^{\varepsilon}_{k}a\rVert_{q}^{q}\Big{)}^{1/q},\qquad
a\in\ell^{q}(\Pi).$ (3.13)

with constants depending only on the minimal index $p_{0}$ and $\varepsilon$,
but not on $p$ and $q$.

An application of Lemma 3.6 on (3.13) yields the claim. The corresponding
statement for $q=\infty$ follows similarly.

The technical part of the proof consists of precise estimates for the Schur-
type norms

$\displaystyle V^{\varepsilon,p}_{j,k}$
$\displaystyle:=\lVert[A,\varphi^{\varepsilon}_{k}]\varphi^{\varepsilon}_{j}\rVert_{S-p},$
$\displaystyle\qquad 0<p\leq 1,\ j,k$ $\displaystyle\in\mathbb{Z}^{d}$ (3.14)
and $\displaystyle V^{\varepsilon}_{j,k}$
$\displaystyle:=\lVert[A,\varphi^{\varepsilon}_{k}]\varphi^{\varepsilon}_{j}\rVert_{\operatorname{Schur}},$
$\displaystyle\qquad j,k$ $\displaystyle\in\mathbb{Z}^{d}.$ (3.15)

of the commutator
$[A,\varphi^{\varepsilon}_{k}]=A\varphi^{\varepsilon}_{k}-\varphi^{\varepsilon}_{k}A$,
when $A\in\mathcal{C}^{p_{0}}(\Lambda,\Pi)$ and $\varphi^{\varepsilon}_{k}$ is
considered as a multiplication operator.

######  Lemma 3.8.

Let $\varepsilon>0$, $p_{0}\in(0,1]$, $p,q\in(p_{0},\infty]$ be such that
$q\leq p$, $\Lambda,\Pi\subset\mathbb{R}^{d}$ be relatively separated and
suppose that $A=(a_{\lambda,\rho})_{\lambda\in\Lambda,\rho\in\Pi}\in
C^{p_{0}}(\Lambda,\Pi)$. Also let $V^{\varepsilon,p}_{j,k}$ and
$V^{\varepsilon}_{j,k}$ be the Schur norms given by (3.14) and (3.15),
$K:=\max_{x}\Phi^{\varepsilon}(x)^{-\min(1,p)}$. Assume that

$\displaystyle\lVert c\rVert_{p}\leq\lVert Ac\rVert_{p},\qquad
c\in\ell^{p}(\Pi).$

Then the following are true:

  1. (i)

If $p\leq 1$, then

$\lVert\varphi^{\varepsilon}_{k}c\rVert_{p}^{q}\leq\lVert\varphi^{\varepsilon}_{k}Ac\rVert_{p}^{q}+K^{q/p}\sum_{j\in\mathbb{Z}^{d}}(V^{\varepsilon,p}_{j,k})^{q/p}\lVert\varphi^{\varepsilon}_{j}c\rVert_{p}^{q},\qquad
c\in\ell^{p}(\Pi)\,.$ (3.16)

  2. (ii)

If $p>1$, then

$\lVert\varphi^{\varepsilon}_{k}c\rVert_{p}\leq\lVert\varphi^{\varepsilon}_{k}Ac\rVert_{p}+K\sum_{j\in\mathbb{Z}^{d}}V^{\varepsilon}_{j,k}\,\lVert\varphi^{\varepsilon}_{j}c\rVert_{p},\qquad
c\in\ell^{p}(\Pi)\,.$ (3.17)

###### Proof.

(i) We apply the triangle inequality for $p\leq 1$ (Lemma 2.1(ii)) and obtain

$\displaystyle\lVert\varphi^{\varepsilon}_{k}c\rVert_{p}^{p}$
$\displaystyle\leq\lVert
A\varphi^{\varepsilon}_{k}c\rVert_{p}^{p}\leq\lVert\varphi^{\varepsilon}_{k}Ac\rVert_{p}^{p}+\lVert[A,\varphi^{\varepsilon}_{k}]c\rVert_{p}^{p}$
$\displaystyle\leq\lVert\varphi^{\varepsilon}_{k}Ac\rVert_{p}^{p}+\sum_{j\in\mathbb{Z}^{d}}\lVert[A,\varphi^{\varepsilon}_{k}]\varphi^{\varepsilon}_{j}(\Phi^{\varepsilon})^{-1}\varphi^{\varepsilon}_{j}c\rVert_{p}^{p}$
$\displaystyle\leq\lVert\varphi^{\varepsilon}_{k}Ac\rVert_{p}^{p}+K\sum_{j\in\mathbb{Z}^{d}}V^{\varepsilon,p}_{j,k}\lVert\varphi^{\varepsilon}_{j}c\rVert_{p}^{p}\,.$

Claim (i) now follows by raising this inequality to the power $q/p\leq 1$ and
applying Lemma 2.1(i).

Assertion (ii) was proved in [22], Equ. (36) with the same argument.

Next we consider the matrix $V^{\varepsilon}$ with entries
$(V^{\varepsilon,p}_{j,k})^{q/p},j,k\in\mathbb{Z}^{d}$ in case $p\leq 1$ and
estimate its $q/p$-Schur norm as $\varepsilon\to 0+$. First we prove the
convergence of the entries of $V^{\varepsilon}$.

######  Lemma 3.9.

Suppose that the hypothesis of Lemma 3.8 hold true. Then for
$\varepsilon\longrightarrow 0^{+}$

$\displaystyle\sup_{j,k\in\mathbb{Z}^{d}}V^{\varepsilon,p}_{j,k}\longrightarrow
0\text{ if }p\leq 1\quad\text{ and
}\quad\sup_{j,k\in\mathbb{Z}^{d}}V^{\varepsilon}_{j,k}\longrightarrow 0\text{
if }p>1.$ (3.18)

###### Proof.

The case $p\geq 1$ of (3.18) was proved in (38) of [22]. The necessary
adaptions for the proof of case $p\leq 1$ are as follows. We first note that
the matrix entries of
$[A,\varphi^{\varepsilon}_{k}]\varphi^{\varepsilon}_{j}$, for
$j,k\in\mathbb{Z}^{d}$, are

$\displaystyle([A,\varphi^{\varepsilon}_{k}]\varphi^{\varepsilon}_{j})_{\lambda,\rho}=-a_{\lambda,\rho}\varphi^{\varepsilon}_{j}(\rho)(\varphi^{\varepsilon}_{k}(\lambda)-\varphi^{\varepsilon}_{k}(\rho)),\qquad\rho\in\Pi,\lambda\in\Lambda.$

Using an envelope $H\in W(C_{b},L^{p_{0}})$ of $A$ and estimate (3.10) for
$\varphi_{k}^{\varepsilon}$, we bound the entries of the commutator by

$\displaystyle\lvert([A,\varphi^{\varepsilon}_{k}]\varphi^{\varepsilon}_{j})_{\lambda,\rho}\rvert^{p}\lesssim
H(\lambda-\rho)^{p}\min\\{1,\varepsilon|\lambda-\rho|\\}^{p}\,.$

Hence, if we define $H^{\varepsilon,p}(x):=H(x)^{p}\min\\{1,\varepsilon\lvert
x\rvert\\}^{p}$, then by choice of $H$,

$\displaystyle
V^{\varepsilon,p}_{j,k}=\lVert[A,\varphi^{\varepsilon}_{k}]\varphi^{\varepsilon}_{j}\rVert_{S-p}\lesssim\mathop{\mathrm{rel}}(\Lambda)^{p/p_{0}}\lVert
H^{\varepsilon,p}\rVert_{W(C_{b},L^{p_{0}/p})}.$

Since $H\in W(C_{b},L^{p_{0}})$ and $p_{0}\leq p\leq 1$, it follows with
dominated convergence that $\lVert
H^{\varepsilon,p}\rVert_{W(C_{b},L^{p_{0}/p})}\longrightarrow 0$, as
$\varepsilon\longrightarrow 0^{+}$. This proves (3.18) for the case $p\leq 1$.

Next we shall estimate $V^{\varepsilon,p}_{j,k}$ in terms of

$\displaystyle\triangle^{\varepsilon,q}(s):=\sum_{t\in{\mathbb{Z}^{d}}:\lvert\varepsilon
t-s\rvert_{\infty}\leq 5}\sup_{z\in[0,1]^{d}+t}\lvert H(z)\rvert^{q}\,,\qquad
s\in\mathbb{Z}^{d}.$ (3.19)

First we have the following.

######  Lemma 3.10.

Let $0<p_{0}\leq q\leq 1$, $H\in W(C_{b},L^{p_{0}})$ and
$\triangle^{\varepsilon,q}(s)$ be given by (3.19). Then

$\displaystyle\sum_{s\in{\mathbb{Z}^{d}},\lvert
s\rvert>6\sqrt{d}}\triangle^{\varepsilon,q}(s)\longrightarrow 0,\mbox{ as
}\varepsilon\longrightarrow 0^{+}.$ (3.20)

###### Proof.

Since $H\in W(L^{\infty},L^{p_{0}})$, we obtain

$\displaystyle\sum_{s\in{\mathbb{Z}^{d}},\lvert
s\rvert>6\sqrt{d}}\triangle^{\varepsilon,q}(s)\leq\sum_{s\in{\mathbb{Z}^{d}},\lvert
s\rvert_{\infty}>6}\triangle^{\varepsilon,q}(s)\lesssim\sum_{t\in{\mathbb{Z}^{d}},\lvert
t\rvert_{\infty}>1/\varepsilon}\sup_{z\in[0,1]^{d}+t}\lvert
H(z)^{q}\rvert\longrightarrow 0,$

as $\varepsilon\longrightarrow 0^{+}$.

######  Lemma 3.11.

Suppose that the hypothesis of Lemma 3.8 hold true with $q\leq 1$ and
$\varepsilon\leq 1$, and let $\triangle^{\varepsilon,q}(s)$ be given by
(3.19). Then for $\lvert k-j\rvert>4$ we have

$\displaystyle(V^{\varepsilon,p}_{j,k})^{q/p}$
$\displaystyle\leq\sup_{\rho\in\Pi}\sum_{\lambda\in\Lambda}\lvert([A,\varphi^{\varepsilon}_{k}]\varphi^{\varepsilon}_{j})_{\lambda,\rho}\rvert^{q}\lesssim\triangle^{\varepsilon,q}(k-j)\text{
if }p\leq 1,$ (3.21) $\displaystyle(V^{\varepsilon}_{j,k})^{q}$
$\displaystyle\lesssim\triangle^{\varepsilon,q}(k-j)\text{ if }p>1.$ (3.22)

###### Proof.

Suppose that $\lvert k-j\rvert>4$. Since $\varphi$ is supported in $B_{2}(0)$,
it follows that
$\varphi^{\varepsilon}_{j}(\rho)\varphi^{\varepsilon}_{k}(\rho)=0$.

As a consequence, the matrix entries of
$[A,\varphi^{\varepsilon}_{k}]\varphi^{\varepsilon}_{j}$ simplify into

$\displaystyle([A,\varphi^{\varepsilon}_{k}]\varphi^{\varepsilon}_{j})_{\lambda,\rho}=-a_{\lambda,\rho}\varphi^{\varepsilon}_{j}(\rho)\varphi^{\varepsilon}_{k}(\lambda),\qquad\rho\in\Pi,\lambda\in\Lambda\,.$

This gives

$\displaystyle\lvert([A,\varphi^{\varepsilon}_{k}]\varphi^{\varepsilon}_{j})_{\lambda,\rho}\rvert^{q}\leq\lvert
H(\lambda-\rho)\rvert^{q}\varphi^{\varepsilon}_{j}(\rho)^{q}\varphi^{\varepsilon}_{k}(\lambda)^{q}.$

Consequently, using (3.4) for $\lvert k-j\rvert>4$ and $q/p\leq 1$, we have

$\displaystyle(V^{\varepsilon,p}_{j,k})^{q/p}$
$\displaystyle=\lVert[A,\varphi_{k}^{\varepsilon}]\varphi_{j}^{\varepsilon}\rVert_{S-p}^{q/p}\leq\sup_{\rho\in\Pi}\sum_{\lambda\in\Lambda}\lvert([A,\varphi^{\varepsilon}_{k}]\varphi^{\varepsilon}_{j})_{\lambda,\rho}\rvert^{q}\leq\sup_{\rho\in\Pi}\sum_{\lambda\in\Lambda}\lvert
H(\lambda-\rho)\rvert^{q}\varphi^{\varepsilon}_{j}(\rho)^{q}\varphi^{\varepsilon}_{k}(\lambda)^{q}\,.$

If $\varphi^{\varepsilon}_{j}(\rho)\varphi^{\varepsilon}_{k}(\lambda)\not=0$,
then $\lvert\varepsilon\rho-j\rvert\leq 2$ and
$\lvert\varepsilon\lambda-k\rvert\leq 2$, whence

$\displaystyle\lvert\varepsilon(\lambda-\rho)+(j-k)\rvert\leq 4.$ (3.23)

Hence

$\displaystyle\sup_{\rho\in\Pi}\sum_{\lambda\in\Lambda}\lvert([A,\varphi^{\varepsilon}_{k}]\varphi^{\varepsilon}_{j})_{\lambda,\rho}\rvert^{q}\lesssim\sup_{\rho\in\Pi}\sum_{\stackrel{{\scriptstyle\lambda\in\Lambda}}{{\lvert\varepsilon(\lambda-\rho)+(j-k)\rvert\leq
4}}}\lvert H(\rho-\lambda)\rvert^{q}.$ (3.24)

For fixed $\varepsilon\leq 1$ we bound the sum in (3.24) by

$\displaystyle\sum_{\lambda\in\Lambda:\lvert\varepsilon(\lambda-\rho)+(j-k)\rvert\leq
4}\lvert
H(\rho-\lambda)\rvert^{q}\leq\sum_{t\in{\mathbb{Z}^{d}}}\sum_{\stackrel{{\scriptstyle\lambda\in\Lambda:\lvert\varepsilon(\lambda-\rho)+(j-k)\rvert\leq
4}}{{(\lambda-\rho)\in[0,1]^{d}+t}}}\lvert H(\rho-\lambda)\rvert^{q}$
$\displaystyle\quad\lesssim\mathop{\mathrm{rel}}(\rho-\Lambda)\sum_{t\in{\mathbb{Z}^{d}}:\lvert\varepsilon
t+(j-k)\rvert_{\infty}\leq 5}\sup_{z\in[0,1]^{d}+t}\lvert H(z)\rvert^{q}$
$\displaystyle\quad\lesssim\triangle^{\varepsilon,q}(k-j)\,,$

since $\mathop{\mathrm{rel}}(\Lambda)$ is translation-invariant. Substituting
this bound in (3.24), we obtain (3.21). Analogously we obtain (3.22).

######  Lemma 3.12.

Suppose that the hypotheses of Lemma 3.8 hold for $p\leq 1$. Then

$\displaystyle\sup_{k\in\mathbb{Z}^{d}}\sum_{j\in\mathbb{Z}^{d}}(V^{\varepsilon,p}_{j,k})^{q/p}+\sup_{j\in\mathbb{Z}^{d}}\sum_{k\in\mathbb{Z}^{d}}(V^{\varepsilon,p}_{j,k})^{q/p}\longrightarrow
0,\mbox{ as }\varepsilon\longrightarrow 0^{+}.$ (3.25)

###### Proof.

Let $\varepsilon\leq 1$ and $\triangle^{\varepsilon,q}$ be defined as in
(3.19). Fix $j\in\mathbb{Z}^{d}$ and use Lemma 3.11 to estimate

$\displaystyle\sum_{k:\lvert
k-j\rvert>6\sqrt{d}}(V^{\varepsilon,p}_{j,k})^{q/p}\lesssim\sum_{k:\lvert
k-j\rvert>6\sqrt{d}}\triangle^{\varepsilon,q}(k-j).$

For the sum over $\\{k\in\mathbb{Z}^{d}:\lvert j-k\rvert\leq 6\sqrt{d}\\}$ we
use the bound

$\displaystyle\sum_{k:\lvert j-k\rvert\leq
6\sqrt{d}}(V^{\varepsilon,p}_{j,k})^{q/p}\leq\\#\\{k:\lvert j-k\rvert\leq
6\sqrt{d}\\}\sup_{s,t}(V^{\varepsilon,p}_{s,t})^{q/p}\lesssim\sup_{s,t}(V^{\varepsilon,p}_{s,t})^{q/p}.$

Hence,

$\displaystyle\sum_{k\in\mathbb{Z}^{d}}(V^{\varepsilon,p}_{j,k})^{q/p}\lesssim\sup_{s,t}(V^{\varepsilon,p}_{s,t})^{q/p}+\sum_{\lvert
s\rvert>6\sqrt{d}}\triangle^{\varepsilon,q}(s),$

which tends to 0 uniformly in $j$ as $\varepsilon\rightarrow 0^{+}$ by Lemmas
3.9 and 3.10. The convergence of the first term in (3.25) follows in exactly
the same way by interchanging the roles of $j$ and $k$.

With the auxiliary results at hand, we now prove Theorem 3.5.

###### Proof of Theorem 3.5.

As already mentioned we restrict ourselves to the case $p_{0}<1$, cf. Remark
3.1. Since $W(C^{0};L^{p_{0}})(\mathbb{R}^{d})\subseteq
W(L^{\infty};L^{1})(\mathbb{R}^{d})$ an application of Proposition 8.1 of [22]
provides the claim in the Banach space case $p,q\geq 1$.

Case $p\leq 1$ and $q<p$ so that $q/p<1$. After multiplying $A$ with a
constant, we may assume that

$\displaystyle\lVert c\rVert_{p}\leq\lVert Ac\rVert_{p},\qquad
c\in\ell^{p}(\Pi),$

since $A$ is bounded from below on $\ell^{p}(\Pi)$ by assumption (3.7). By
Lemma 3.8, (3.16), we obtain

$\sum_{k\in\mathbb{Z}^{d}}\lVert\varphi^{\varepsilon}_{k}c\rVert_{p}^{q}\leq\sum_{k\in\mathbb{Z}^{d}}\lVert\varphi^{\varepsilon}_{k}Ac\rVert_{p}^{q}+K^{q/p}\sum_{k\in\mathbb{Z}^{d}}\sum_{j\in\mathbb{Z}^{d}}(V^{\varepsilon,p}_{j,k})^{q/p}\lVert\varphi^{\varepsilon}_{j}c\rVert_{p}^{q},\quad
c\in\ell^{p}(\Pi)\,,$ (3.26)

for some $K>0$. According to Lemma 3.12 we may choose $\varepsilon>0$ such
that

$K^{q/p}\sup_{j\in\mathbb{Z}^{d}}\sum_{k\in\mathbb{Z}^{d}}(V^{\varepsilon,p}_{j,k})^{q/p}<\frac{1}{2}\,.$

Using this bound in (3.26) and Proposition 3.2, we obtain that

$\displaystyle\sum_{k\in\mathbb{Z}^{d}}\lVert\varphi^{\varepsilon}_{k}c\rVert_{p}^{q}\leq\sum_{k\in\mathbb{Z}^{d}}\lVert\varphi^{\varepsilon}_{k}Ac\rVert_{p}^{q}+1/2\sum_{k\in\mathbb{Z}^{d}}\lVert\varphi^{\varepsilon}_{k}c\rVert_{p}^{q}\,.$

Hence,

$\displaystyle\left(\sum_{k\in\mathbb{Z}^{d}}\lVert\varphi^{\varepsilon}_{k}c\rVert_{p}^{q}\right)^{1/q}\leq
2^{1/q}\left(\sum_{k\in\mathbb{Z}^{d}}\lVert\varphi^{\varepsilon}_{k}Ac\rVert_{p}^{q}\right)^{1/q}.$
(3.27)

Using the equivalent norm of Lemma 3.7 in (3.27), we deduce that for all
$p_{0}\leq q<p$

$\displaystyle\lVert c\rVert_{q}\lesssim\lVert Ac\rVert_{q},$

with a constant independent of $q$ (since $2^{1/q}\leq 2^{1/p_{0}}$). This
completes the proof of the case $p_{0}\leq q\leq p$.

Case $p<1$ and $p<q$. Again we may assume that

$\displaystyle\lVert c\rVert_{p}\leq\lVert Ac\rVert_{p},\quad
c\in\ell^{p}(\Pi).$

According to Lemma 3.12 in case $q=p$ we may choose $\varepsilon>0$ such that

$\displaystyle
K\sup_{j\in\mathbb{Z}^{d}}\sum_{k\in\mathbb{Z}^{d}}V^{\varepsilon,p}_{j,k}<\frac{1}{2}\quad\text{
and }\quad
K\sup_{k\in\mathbb{Z}^{d}}\sum_{j\in\mathbb{Z}^{d}}V^{\varepsilon,p}_{j,k}<\frac{1}{2}\,.$
(3.28)

Due to Lemma 3.8 we have, for $c\in\ell^{p}(\Pi)$,

$\displaystyle\lVert\varphi^{\varepsilon}_{k}c\rVert_{p}^{p}\leq\lVert\varphi^{\varepsilon}_{k}Ac\rVert_{p}^{p}+K\sum_{j\in\mathbb{Z}^{d}}V^{\varepsilon,p}_{j,k}\lVert\varphi^{\varepsilon}_{j}c\rVert_{p}^{p}.$

We set $a^{\varepsilon}_{k}=\lVert\varphi_{k}^{\varepsilon}c\rVert_{p}^{p}$,
$b^{\varepsilon}_{k}=\lVert\varphi_{k}^{\varepsilon}Ac\rVert_{p}^{p}$ and let
$V^{\varepsilon}$ be the operator associated to the matrix
$V^{\varepsilon,p}_{j,k}$. Then the previous inequality can be written as

$\displaystyle a^{\varepsilon}_{k}\leq
b^{\varepsilon}_{k}+K(V^{\varepsilon}a^{\varepsilon})_{k}.$

We take the $q/p-$norm of the previous estimate, apply the triangle inequality
first. Next we apply Proposition 3.1, use (3.28) and get

$\displaystyle\lVert a^{\varepsilon}\rVert_{q/p}\leq\lVert
b^{\varepsilon}\rVert_{q/p}+K\lVert
V^{\varepsilon}a^{\varepsilon}\rVert_{q/p}\leq\lVert
b^{\varepsilon}\rVert_{q/p}+\frac{1}{2}\lVert a^{\varepsilon}\rVert_{q/p}.$

In other words we have $\lVert a^{\varepsilon}\rVert_{q/p}\leq 2\lVert
b^{\varepsilon}\rVert_{q/p}$. Undoing the abbreviations, this means that

$\displaystyle\left(\sum_{k\in\mathbb{Z}^{d}}\lVert\varphi^{\varepsilon}_{k}c\rVert_{p}^{q}\right)^{1/q}=\lVert
a^{\varepsilon}\rVert_{p}^{1/p}\leq 2^{1/p}\lVert
b^{\varepsilon}\rVert_{p}^{1/p}=2^{1/p}\left(\sum_{k\in\mathbb{Z}^{d}}\lVert\varphi^{\varepsilon}_{k}Ac\rVert_{p}^{q}\right)^{1/q}.$

An application of the norm equivalence of Lemma 3.7 provides the claim

$\displaystyle\lVert c\rVert_{q}\lesssim\lVert Ac\rVert_{q}.$

As observed in [22, Rem. A.2] the lower bound guaranteed by Theorem 3.5 is
uniform for all $p$.

As an immediate consequence of Theorem 3.5 we get a first spectral invariance
result of a matrix $A\in\mathcal{C}^{p_{0}}(\Lambda,\Pi)$ by means of standard
functional analytical arguments.

######  Corollary 3.13.

Let $p_{0}\in(0,1]$, $\Lambda,\Pi\subseteq\mathbb{R}^{d}$ be relatively
separated, and suppose that $A\in C^{p_{0}}(\Lambda,\Pi)$ is invertible from
$\ell^{p}(\Pi)$ to $\ell^{p}(\Lambda)$ for some $p\in[p_{0},\infty]$. Then the
following is true:

  1. (i)

$A$ is injective from $\ell^{q}(\Pi)$ to $\ell^{q}(\Lambda)$ for all
$q\in[p_{0},\infty]$.

  2. (ii)

$A$ is invertible from $\ell^{q}(\Pi)$ to $\ell^{q}(\Lambda)$ for all
$q\in[p,\infty)$.

###### Proof.

As already observed, $A:\ell^{p}(\Pi)\to\ell^{p}(\Lambda)$ satisfies the
stability condition $\|c\|_{p}\lesssim\|Ac\|_{p}$, if and only if $A$ is one-
to-one on $\ell^{p}(\Pi)$ and has closed range in $\ell^{p}(\Lambda)$. Thus,
if $A$ is invertible on some $\ell^{p}(\Pi)$, then by Theorem 3.5 $A$
satisfies the stability condition (3.7) for all $q\geq p_{0}$. Consequently
$A$ is one-to-one on all $\ell^{q}(\Pi)$ for $q\geq p_{0}$, which is (i).

If $q\geq p$, then $\ell^{p}(\Lambda)$ is dense in $\ell^{q}(\Lambda)$. Since
$A$ is onto $\ell^{p}(\Lambda)$, we obtain
$\ell^{p}(\Lambda)=A\ell^{p}(\Pi)\subseteq\ell^{q}(\Lambda)$, and $A$ has also
dense range in $\ell^{q}(\Lambda)$. Consequently $A$ is onto
$\ell^{q}(\Lambda)$ and thus invertible on $\ell^{q}(\Lambda)$.

In the case, when $A$ in invertible is the Hilbert space $\ell^{2}$, the above
results are already contained in [34, Theorem 4.6 and Theorem 8.5] and in [8,
Theorem 3.9] and in [10].

Note that Corollary 3.13 asserts only that the invertibility of $A$ on
$\ell^{p}$ implies the invertibility on the larger space $\ell^{q}$, $q>p$. To
obtain the same conclusion for the smaller spaces $\ell^{q},q<p$, we need to
refine our arguments. We will show that the inverse of $A$ has an envelope
belonging to the same Wiener amalgam space than the envelope of $A$. The main
idea of the proof of the following Theorem is taken from [38].

######  Theorem 3.14.

Let $0<p_{0}\leq 1$, $\Lambda,\Pi\subseteq\mathbb{R}^{d}$ be relatively
separated. Suppose $A\in\mathcal{C}^{p_{0}}(\Lambda,\Pi)$ is invertible from
$\ell^{p}(\Pi)$ to $\ell^{p}(\Lambda)$ for some $p\in[p_{0},\infty]$. Then
$A^{-1}\in\mathcal{C}^{p_{0}}(\Pi,\Lambda)$.

###### Proof.

By assumption the matrix $A=(a_{\lambda,\rho})_{\lambda\in\Lambda,\rho\in\Pi}$
has an envelope $H_{1}\in W(C_{b},L^{p_{0}})$ so that $|a_{\lambda,\rho}|\leq
H_{1}(\lambda-\rho)$. We need to prove that the matrix $A^{-1}$ also has an
envelope $H_{2}\in W(C_{b},L^{p_{0}})$. For this we recall the notation used
in the previous lemmas: $K=\max_{x}\Phi^{\varepsilon}(x)^{-\min(1,p)}$ from
Lemma 3.8, and

$\widetilde{V}^{\varepsilon}=\begin{cases}\big{(}(KV^{\varepsilon,p}_{j,k})^{p_{0}/p}\big{)}_{j,k\in\mathbb{Z}^{d}},&p\leq
1,\\\\[4.30554pt]
\big{(}(KV_{j,k}^{\varepsilon})^{p_{0}}\big{)}_{j,k\in\mathbb{Z}^{d}},&p>1,\end{cases}$

where $V^{\varepsilon,p}_{j,k}$ and $V^{\varepsilon}_{j,k}$ are defined as in
(3.14) and (3.15).

Given $c\in\ell^{p}(\Pi)$, the sequences $a$ and $a_{A}$ are defined by

$a_{k}=\|\varphi^{\varepsilon}_{k}c\|_{p}^{p_{0}}\quad\text{and}\quad
a_{A,k}=\|\varphi^{\varepsilon}_{k}Ac\|_{p}^{p_{0}},\qquad
k\in\mathbb{Z}^{d}.$

With this notation Lemma 3.8 (and additionally Lemma 2.1 if $p>1$) says that

$a\leq a_{A}+\widetilde{V}^{\varepsilon}a\,.$ (3.29)

Furthermore, Lemma 3.11 can be reformulated as saying that
$\widetilde{V}^{\varepsilon}$ is convolution-dominated. In fact, recall from
(3.19) that

$\displaystyle\Delta^{\varepsilon,p_{0}}(s):=\sum_{t\in\mathbb{Z}^{d}:|\varepsilon
t-s|_{\infty}\leq 5}\sup_{z\in[0,1]^{d}+t}|H_{1}(z)|^{p_{0}}\,,$

and define $\Psi^{\varepsilon}$ by
$\Psi^{\varepsilon}(s):=\Delta^{\varepsilon,p_{0}}(s)$ for all
$s\in\mathbb{Z}^{d}$ with $|s|>6\sqrt{d}$ and for all $s\in\mathbb{Z}^{d}$
with $|s|\leq 6\sqrt{d}$ by
$\Psi^{\varepsilon}(s):=\left(\sup_{j,k\in\mathbb{Z}^{d}}|V_{j,k}^{\varepsilon,p}|\right)^{p_{0}/p}$
if $p\leq 1$ and by
$\Psi^{\varepsilon}(s):=\left(\sup_{j,k\in\mathbb{Z}^{d}}|V_{j,k}^{\varepsilon}|\right)^{p_{0}}$
if $p\geq 1$. In view of the normalization $K$, $\tilde{V}^{\varepsilon}$
possesses the envelope $K^{p_{0}/\min(1,p)}\Psi^{\varepsilon}$. This means
that

$\displaystyle(V^{\varepsilon,p}_{j,k})^{p_{0}/p}$
$\displaystyle\lesssim\Psi^{\varepsilon}(k-j),$ $\displaystyle\qquad p$
$\displaystyle\leq 1,\ j,k\in\mathbb{Z}^{d}\,,$ (3.30) and $\displaystyle
V^{\varepsilon}_{j,k}$ $\displaystyle\lesssim\Psi^{\varepsilon}(k-j),$
$\displaystyle\qquad p$ $\displaystyle>1,\ j,k\in\mathbb{Z}^{d}\,.$ (3.31)

Our next goal is to represent $(I-\widetilde{V}^{\varepsilon})^{-1}$ as a
Neumann series. For this we choose $\varepsilon>0$ such that

$\|\widetilde{V}^{\varepsilon}\|_{S-1}\leq
K^{p_{0}/\min(1,p)}\|\Psi^{\varepsilon}\|_{1}\leq\frac{1}{2}.\,$ (3.32)

This is possible due to Lemma 3.10 and Lemma 3.9, since
$\sharp\\{s\in\mathbb{Z}^{d}:|s|\leq 6\sqrt{d}\\}$ is finite and depends only
on the dimension $d$.

As a consequence the geometric series
$\widetilde{W}:=\sum_{k=1}^{\infty}\left(\widetilde{V}^{\varepsilon}\right)^{k}$
converges in the $\|.\|_{S-1}$-norm and we obtain

$(\mathrm{I}-\widetilde{V}^{\varepsilon})^{-1}=\sum_{k=0}^{\infty}\left(\widetilde{V}^{\varepsilon}\right)^{k}=\mathrm{I}+\widetilde{W}\,.$

Since all entries of $\widetilde{V}^{\varepsilon}$ are non-negative by
definition, $\widetilde{W}$ also has only non-negative entries and preserves
(pointwise) inequalities. Moreover, since $\widetilde{V}^{\varepsilon}$ is
convolution-dominated, so is $\widetilde{W}$, and by (3.32) there exists an
envelope $W\in\ell^{1}(\mathbb{Z}^{d})$ such that

$\widetilde{W}_{jk}\leq W(k-j)\qquad j,k\in\mathbb{Z}^{d}\,.$

Now (3.29) yields

$a=(\mathrm{I}+\widetilde{W})(\mathrm{I}-\widetilde{V}^{\varepsilon})a\leq(\mathrm{I}+\widetilde{W})a_{A}\,,$
(3.33)

or entrywise

$\|\varphi_{k}^{\varepsilon}c\|_{p}^{p_{0}}\leq\|\varphi_{k}^{\varepsilon}Ac\|_{p}^{p_{0}}+\sum_{j\in\mathbb{Z}^{d}}W(k-j)\|\varphi_{j}^{\varepsilon}Ac\|_{p}^{p_{0}}\,.$
(3.34)

Since $A$ is assumed to be invertible as a map from $\ell^{p}(\Pi)$ to
$\ell^{p}(\Lambda)$, there exist $b_{\lambda}\in\ell^{p}(\Pi)$ such that
$Ab_{\lambda}=\delta_{\lambda}$, whence the matrix $B$ with entries
$b_{\rho\lambda}=(b_{\lambda})_{\rho}$ is the inverse of $A$. Using
$b_{\lambda}$ in (3.34) we obtain

$\|\varphi_{k}^{\varepsilon}b_{\lambda}\|_{p}^{p_{0}}\leq\|\varphi_{k}^{\varepsilon}\delta_{\lambda}\|_{p}^{p_{0}}+\sum_{j\in\mathbb{Z}^{d}}W(k-j)\|\varphi_{j}^{\varepsilon}\delta_{\lambda}\|_{p}^{p_{0}}\,.$
(3.35)

Note that

$\|\varphi_{k}^{\varepsilon}\delta_{\lambda}\|_{p}^{p_{0}}=\big{(}\sum_{\rho\in\Lambda}\varphi(\varepsilon\rho-k)^{p}\delta_{\lambda}(\rho)\big{)}^{p_{0}/p}=\varphi(\varepsilon\lambda-k)^{p_{0}}\,.$

Let $\lambda\in\Lambda$, $\rho\in\Pi$. For the off-diagonal decay it suffices
to consider only indices satisfying $\varepsilon|\lambda-\rho|>4$. Choose
$k_{\rho}\in\mathbb{Z}^{d}$ such that

$|\varepsilon\rho-k_{\rho}|<2\quad\text{ and }\quad\varphi(\varepsilon\rho-
k_{\rho})^{p_{0}}\geq c$

for some constant $c$ (in fact, by (A.2) in the proof of Lemma 3.6
$c=\eta^{-1}$). Then

$|\varepsilon\lambda-k_{\rho}|\geq\varepsilon|\lambda-\rho|-|\varepsilon\rho-
k_{\rho}|>2\,,$

and consequently $\varphi(\varepsilon\lambda-k_{\rho})=0$ in (3.35) since
$\text{supp}(\varphi)\subseteq B_{2}(0)$. Now (3.35) simplifies to

$c|(b_{\lambda})_{\rho}|^{p_{0}}\leq\|\varphi_{k_{\rho}}^{\varepsilon}b_{\lambda}\|_{p}^{p_{0}}\leq\sum_{j\in\mathbb{Z}^{d}}W(k_{\rho}-j)\varphi(\varepsilon\lambda-j)^{p_{0}}\leq\sum_{j:|j-\varepsilon\lambda|<2}W(k_{\rho}-j)\,.$
(3.36)

This inequality suggests the following envelope for $B=A^{-1}$. Let

$H(x)=\sum_{l\in\mathbb{Z}^{d}:|l-\varepsilon x|<4}W(l).$ (3.37)

To obtain a continuous envelope, we use a cut-off function $\psi\in
C^{\infty}_{c}(\mathbb{R}^{d})$ satisfying $0\leq\psi\leq 1$ and $\psi(x)=1$
for $|x|\leq 4$ and set

$\displaystyle\tilde{H}(x)=\sum_{l\in\mathbb{Z}^{d}}W(l)\psi(\varepsilon
x-l)\,.$

Since $\psi$ has compact support and $W(C_{b},\ell^{1})$ is translation
invariant, we have

$\displaystyle\|\tilde{H}\|_{W(C_{b},\ell^{1})}\leq\sum_{l\in\mathbb{Z}^{d}}|W(l)|\sup_{l\in\mathbb{Z}^{d}}\|\psi(\varepsilon\cdot-l)\|_{W(C_{b},\ell^{1})}\lesssim\|W\|_{\ell^{1}}\,,$

therefore $\tilde{H}\in W(C_{b},\ell^{1})$.

Furthermore, since
$|k_{\rho}-j-\varepsilon(\lambda-\rho)|\leq|k_{\rho}-\varepsilon\rho|+|\varepsilon\lambda-j|<4$,
(3.37) says that

$|b_{\rho\lambda}|^{p_{0}}\leq H(\rho-\lambda)\leq\tilde{H}(\rho-\lambda)\,,$
(3.38)

or

$|b_{\rho\lambda}|=|(b_{\lambda})_{\rho}|\leq\tilde{H}(\rho-\lambda)^{1/p_{0}}\,,$

and $\tilde{H}^{1/p_{0}}\in W(C_{b},L^{p_{0}})$, as claimed.

This theorem enables us to extend the spectral invariance result of Corollary
3.13 to the case $p_{0}\leq q\leq p$ as follows.

######  Theorem 3.15.

Let $p_{0}\in(0,1]$, $\Lambda,\Pi\subseteq\mathbb{R}^{d}$ be relatively
separated and $A\in C^{p_{0}}(\Lambda,\Pi)$ be invertible from $\ell^{p}(\Pi)$
to $\ell^{p}(\Lambda)$ for some $p\in[p_{0},\infty]$. Then $A$ is invertible
from $\ell^{q}(\Pi)$ to $\ell^{q}(\Lambda)$ for all $q\in[p_{0},\infty)$.

###### Proof.

According to Corollary 3.13(ii) $A$ is invertible on $\ell^{q}$ for $q\geq p$
and $A$ is one-to-one on all $\ell^{q}$. So it remains to prove that $A$ is
surjective for $p_{0}\leq q<p$. We assume that $A$ is invertible on $\ell^{p}$
with inverse $A^{-1}$. Hence for
$u\in\ell^{q}(\Lambda)\subseteq\ell^{p}(\Lambda)$ there is some
$c\in\ell^{p}(\Pi)$ with $c=A^{-1}u$. By Theorem 3.14 $A^{-1}$ is bounded on
$\ell^{q}$, consequently, since $u\in\ell^{q}$ we have $c=A^{-1}u\in\ell^{q}$.
Thus $A$ is onto $\ell^{q}(\Lambda)$.

The following consequence explains why the statement of Theorem 3.15 is
referred to as the spectral invariance property.

######  Corollary 3.16.

Let $0<p_{0}\leq 1$ be arbitrary. If $A\in\mathcal{C}^{p_{0}}(\Lambda)$ and
$A$ is invertible on $\ell^{p}(\Lambda)$ for some $p\in[p_{0},\infty]$, then
$A^{-1}\in\mathcal{C}^{p_{0}}(\Lambda)$ and

$\displaystyle
Sp_{\mathcal{B}(\ell^{q})}(A)=Sp_{\mathcal{C}^{p_{0}}}(A)\qquad\text{ for all
}q\in[p_{0},\infty),$

where $Sp_{\mathcal{A}}(A)$ denotes the spectrum of $A$ in the algebra
$\mathcal{A}$.

In the literature many variations of this spectral invariance result exist.
For an overview of those variations we refer to [20]. Here we just want to
mention the following ones: In case $p_{0}=1$ and $p=q=2$ the previous theorem
also holds in the weighted case. It was proved by Baskakov e.g. in [3, 4] and
by Sjöstrand [38] in the unweighted case.

##  4\. Spectral Invariance of Pseudodifferential Operators

The aim of this section is to transfer the results on the spectral invariance
of matrices to pseudodifferential operators on unweighted modulation spaces.
In the proofs we mainly follow [19] and [24] and replace an arbitrary
spectrally invariant Banach algebra of matrices by the quasi-Banach algebra
$\mathcal{C}^{p_{0}}$.

First we list up all needed definitions and properties to reach that aim. We
start with recalling the modulation spaces.

###  4.1. Modulation Spaces $M^{p,q}$.

For the definition of those spaces we need the short-time Fourier transform,
which we recall now.

Let $g\in\mathcal{S}({\mathbb{R}^{d}})\setminus\\{0\\}$ be fixed. For every
$f\in\mathcal{S}^{\prime}(\mathbb{R}^{d})$, the short-time Fourier transform
$V_{g}f$ is the function on $\mathbb{R}^{2d}$ defined by the formula

$\displaystyle V_{g}f(x,\xi):=\langle f,g(\cdot-x)e^{2\pi
i\xi\cdot\,\cdot}\rangle.$ (4.1)

Here $\langle\,\cdot\,,\,\cdot\,\rangle$ is the unique extension of the
$L^{2}$ scalar product on
$\mathcal{S}(\mathbb{R}^{d})\times\mathcal{S}(\mathbb{R}^{d})$ into
$\mathcal{S}^{\prime}(\mathbb{R}^{d})\times\mathcal{S}(\mathbb{R}^{d})$. We
observe that if $f\in L^{p}(\mathbb{R}^{d})$ for some $p\in[1,\infty]$, then
$V_{g}f$ is given by (1.1). If $g$ and $f$ are both defined on
$\mathbb{R}^{2d}$, then $V_{g}f$ is a function on $\mathbb{R}^{4d}$.

We recall that for $0<p,q\leq\infty$ and fixed
$g\in\mathcal{S}({\mathbb{R}^{d}})\setminus\\{0\\}$, the _modulation space_
$M^{p,q}(\mathbb{R}^{d})$ consists of all
$f\in\mathcal{S}^{\prime}(\mathbb{R}^{d})$ such that (1.2) is finite. If
$1\leq p,q\leq\infty$, then $M^{p,q}(\mathbb{R}^{d})$ is a Banach space with
norm $\|\,\cdot\,\|_{M^{p,q}}$. Otherwise $M^{p,q}(\mathbb{R}^{d})$ is a
quasi-Banach space with quasi-norm $\|\,\cdot\,\|_{M^{p,q}}$. We write
$M^{p}=M^{p,p}$.

It is well-known that the definition of $M^{p,q}(\mathbb{R}^{d})$ is
independent of the choice of the window function
$g\in\mathcal{S}({\mathbb{R}^{d}})$ [18]. For $p<1$ or $q<1$ the proof of this
fact can be found in [17]. Additionally, the Schwartz space
$\mathcal{S}({\mathbb{R}^{d}})$ is dense in $M^{p,q}(\mathbb{R}^{d})$ in the
case $p,q<\infty$, cf. [17, Remark 14].

Due to [33, Theorem 3.6] and [17, Theorem 3.4.] the following continuous
embeddings of modulation spaces hold.

######  Proposition 4.1.

Let $0<p_{1}\leq p_{2}\leq\infty$ and $0<q_{1}\leq q_{2}\leq\infty$. Then

$\mathcal{S}(\mathbb{R}^{d})\hookrightarrow
M^{p_{1},q_{1}}(\mathbb{R}^{d})\hookrightarrow
M^{p_{2},q_{2}}(\mathbb{R}^{d})\hookrightarrow\mathcal{S}^{\prime}(\mathbb{R}^{d}).$

###  4.2. Gabor frames

For the definition of Gabor frames it is convenient to use time-frequency
shifts $\pi(z)f$ of $f\in\mathcal{S}^{\prime}(\mathbb{R}^{d})$ given by

$\displaystyle(\pi(z)f)(t):=e^{2\pi i\xi\cdot t}f(t-x),\qquad
z=(x,\xi)\in\mathbb{R}^{2d},\ t\in\mathbb{R}^{d}.$

The _Gabor system_ with respect to the _(Gabor) atom_ $g\in
M^{1}(\mathbb{R}^{d})\backslash\\{0\\}$ and lattice
$\Lambda\subseteq\mathbb{R}^{2d}$ is given by

$\displaystyle\mathcal{G}(g,\Lambda)=\\{\pi(\lambda)g:\lambda\in\Lambda\\}.$

Then the _analysis operator_ and _synthesis operator_

$C_{g}\,:\,M^{\infty}(\mathbb{R}^{d})\to\ell^{\infty}(\Lambda)\quad\text{and}\quad
D_{g}\,:\,\ell^{\infty}(\Lambda)\to M^{\infty}(\mathbb{R}^{d}),$

respectively, with respect to $g$ and $\Lambda$, are given by

$C_{g}f=\big{(}\langle
f,\pi(\lambda)g\rangle\big{)}_{\lambda\in\Lambda}\quad\text{and}\quad
D_{g}c=\sum_{\lambda\in\Lambda}c_{\lambda}\pi(\lambda)g,$

when $f\in M^{\infty}(\mathbb{R}^{d})$ and
$c=(c_{\lambda})_{\lambda\in\Lambda}\in\ell^{\infty}(\Lambda)$. Here the
series converges in $\mathcal{S}^{\prime}(\mathbb{R}^{d})$.

The (Gabor) frame operator
$S=S_{g_{1},g_{2},\Lambda}:M^{\infty}(\mathbb{R}^{d})\to
M^{\infty}(\mathbb{R}^{d})$, $g_{1},g_{2}\in M^{1}(\mathbb{R}^{d})$ is defined
by

$\displaystyle S_{g_{1},g_{2},\Lambda}f:=\sum_{\lambda\in\Lambda}\langle
f,\pi(\lambda)g_{1}\rangle\pi(\lambda)g_{2}.$ (4.2)

If $g_{1}=g_{2}=g$ we write $S_{g,\Lambda}$ instead of
$S_{g_{1},g_{2},\Lambda}$. It follows that $C_{g},D_{g}$ and $S_{g,\Lambda}$
are well-defined and continuous (see e. g. Chapters 11-14 in [18]). Let
$g\in\mathcal{S}({\mathbb{R}^{d}})\backslash\\{0\\}$. If $S_{g,\Lambda}$ is
bounded and invertible on $L^{2}(\mathbb{R}^{d})$, then we call
$\mathcal{G}(g,\Lambda)$ _Gabor frame_. For rectangular lattices
$\Lambda=\alpha\mathbb{Z}^{d}\times\beta\mathbb{Z}^{d}$ the existence of Gabor
frames is well understood, see [9, 18, 31, 45].

For every Gabor frame $\mathcal{G}(g,\Lambda)$ over a lattice, there exists a
dual window $\gamma=S^{-1}g\in L^{2}(\mathbb{R}^{d})$, so that every $f$ can
be expanded into a Gabor expansion

$\displaystyle f$
$\displaystyle=D_{g}C_{\gamma}f=\sum_{\lambda\in\Lambda}\left<f,\pi(\lambda)\gamma\right>\pi(\lambda)g,$
(4.3) $\displaystyle f$
$\displaystyle=D_{\gamma}C_{g}f=\sum_{\lambda\in\Lambda}\left<f,\pi(\lambda)g\right>\pi(\lambda)\gamma.$
(4.4)

For $g\in\mathcal{S}(\mathbb{R}^{d})$ a fundamental result of Janssen [31]
asserts that also $\gamma\in\mathcal{S}(\mathbb{R}^{d})$. Then the expansion
formulas (4.3) and (4.4) hold for every
$f\in\mathcal{S}^{\prime}(\mathbb{R}^{d})$ with weak-$*$-convergence.

A Gabor frame $\mathcal{G}(g,\Lambda)$ is called tight if
$S_{g,\Lambda}=C\,\mathrm{Id}$ for some $C>0$. In this case
$\gamma=S_{g,\Lambda}^{-1}g=C^{-1}g$, and the Gabor expansion

$f=C^{-1}\sum_{\lambda\in\Lambda}\left<f,\pi(\lambda)\gamma\right>\pi(\lambda)g$

looks like an orthonormal expansion.

Tight Gabor frames with the constant $C=1$ can be constructed as follows. Let
$\mathcal{G}(g,\Lambda)$ be a Gabor frame with frame operator $S_{g,\Lambda}$.
Due to [19, Lemma 5.16] $S^{-1/2}_{g,\Lambda}\mathcal{G}(g,\Lambda)$ is a
tight Gabor frame with constant $C=1$. By applying this procedure to a
Gaussian window, one sees that there exist tight Gabor frames
$\mathcal{G}(g,\Lambda)$ with $g\in\mathcal{S}(\mathbb{R}^{d})$.

For future references we remark that if $p\in(0,\infty]$ and
$f\in\mathcal{S}^{\prime}(\mathbb{R}^{d})$, then

$\displaystyle f\in M^{p}(\mathbb{R}^{d})\quad\Leftrightarrow\quad
C_{g}f\in\ell^{p}(\Lambda)\quad\Leftrightarrow\quad
C_{\gamma}f\in\ell^{p}(\Lambda),$ (4.5)

with norm equivalence
$\|f\|_{M^{p}}\asymp\|C_{g}f\|_{\ell^{p}}\asymp\|C_{\gamma}f\|_{\ell^{p}}.$

###  4.3. Pseudodifferential Operators

For a real-valued $d\times d$-matrix $A\in\mathbb{R}^{d\times d}$, and a
symbol $\mathfrak{a}\in\mathcal{S}^{\prime}(\mathbb{R}^{2d})$ the
pseudodifferential operator $Op_{A}(\mathfrak{a})$ is defined by

$\displaystyle
Op_{A}(\mathfrak{a})u(x)=\int\int\mathfrak{a}(x-A(x-y),\xi)f(y)e^{2\pi
i(x-y)\cdot\xi}\,dyd\xi,\qquad u\in\mathcal{S}(\mathbb{R}^{d}),$

where the integrals should be interpreted in distribution sense, if necessary.
If $A=0$, then $Op_{A}(\mathfrak{a})$ agrees with the Kohn-Nirenberg or normal
representation $a(x,D)$. If instead $A=\frac{1}{2}I$, where $I$ is the
$d\times d$ identity matrix, then $Op_{A}(\mathfrak{a})$ is the Weyl
quantization $\mathfrak{a}^{w}$ of $\mathfrak{a}$.

By [29, 44] for each symbol
$\mathfrak{a}_{1}\in\mathcal{S}^{\prime}(\mathbb{R}^{2d})$ and each
$A_{1},A_{2}\in\mathbb{R}^{d\times d}$ there is a unique
$\mathfrak{a}_{2}\in\mathcal{S}^{\prime}(\mathbb{R}^{2d})$ such that
$Op_{A_{1}}(\mathfrak{a}_{1})=Op_{A_{2}}(\mathfrak{a}_{2})$, and that

$\displaystyle
Op_{A_{1}}(\mathfrak{a}_{1})=Op_{A_{2}}(\mathfrak{a}_{2})\Leftrightarrow\mathfrak{a}_{2}(x,\xi)=e^{2\pi
it((A_{1}-A_{2})D_{\xi})\cdot D_{x}}\mathfrak{a}_{1}(x,\xi).$ (4.6)

We refer to [44] for the proof of the following result.

######  Proposition 4.2.

Let $A,A_{1},A_{2}\in\mathbb{R}^{d\times d}$ and $p,q\in(0,\infty]$. Then the
following is true:

  1. (1)

$e^{2\pi it(AD_{\xi})\cdot D_{x}}$ is an isomorphism on
$M^{p,q}(\mathbb{R}^{2d})$;

  2. (2)

$Op_{A_{1}}(M^{p,q}(\mathbb{R}^{2d}))=Op_{A_{2}}(M^{p,q}(\mathbb{R}^{2d}))$.

We can write a Weyl operator by means of the Wigner distribution $W$ of
$f,g\in L^{2}(\mathbb{R}^{d})$ which is defined by

$\displaystyle
W(f,g)(x,\xi):=\int_{\mathbb{R}^{d}}f(x+t/2)\overline{g(x-t/2)}e^{2\pi
i\xi\cdot t}\,dt,\qquad x,\xi\in\mathbb{R}^{d}.$ (4.7)

Denote the inversion ğ of $g\in\mathcal{S}^{\prime}(\mathbb{R}^{d})$ by
$\text{\u{g}}(x):=g(-x)$ for all $x\in\mathbb{R}^{d}$. Then

$\displaystyle W(f,g)(x,\xi)=2^{d}e^{4\pi
ix\cdot\xi}V_{\text{\u{g}}}f(2x,2\xi)\,,$

and the Wigner distribution is just a slight modification of the short-time
Fourier transform. Since the short-time Fourier transform
$V_{.}:\mathcal{S}({\mathbb{R}^{d}})\times\mathcal{S}({\mathbb{R}^{d}})\rightarrow\mathcal{S}(\mathbb{R}^{2d})$
we immediately get $W(f,g)\in\mathcal{S}(\mathbb{R}^{2d})$ for all
$f,g\in\mathcal{S}({\mathbb{R}^{d}})$.

By means of the Wigner distribution the Weyl operator of a symbol
$\mathfrak{a}\in\mathcal{S}^{\prime}({\mathbb{R}^{2d}})$ is given by the
formula

$\displaystyle\left<\mathfrak{a}^{w}f,g\right>=\left<\mathfrak{a},W(g,f)\right>\qquad\text{
for all }f,g\in\mathcal{S}({\mathbb{R}^{d}}).$

Pseudodifferential operators of Weyl form are continuous maps from
$\mathcal{S}({\mathbb{R}^{d}})$ to $\mathcal{S}^{\prime}(\mathbb{R}^{d})$, see
[42] and [29]. Moreover they are continuous as maps between certain modulation
spaces, cf. [18] and [21].

######  Proposition 4.3.

Let $0<p_{0}\leq 1$, $p,q\in[p_{0},\infty]$ and $\mathfrak{a}\in
M^{\infty,p_{0}}({\mathbb{R}^{2d}})$. Then $\mathfrak{a}^{w}$ is bounded on
$M^{p,q}(\mathbb{R}^{d})$.

As proved in [43, Theorem 3.1], this theorem also holds for more general
weighted modulation spaces.

######  Remark 4.1.

By [42] it follows that $M^{p,q}({\mathbb{R}^{2d}})$ is invariant under
actions with chirps $e^{i(AD_{\xi})\cdot D_{x}}$ with $A\in M(d\times
d;\mathbb{C})$ for all $p,q\in(0,\infty]$. Hence all results concerning Weyl
operators of this paper also hold for operators of the form $Op_{A}(a)$.

Next we show that Gabor frame operators with windows in $M^{p_{0}}$ are
pseudo-differential operators with symbols in $M^{\infty,p_{0}}$.

######  Proposition 4.4.

Let $p_{0}\in(0,1]$, $g_{1},g_{2}\in M^{p_{0}}(\mathbb{R}^{d})$, $A\in
M(d\times d;\mathbb{R})$ be a $d\times d$-matrix,
$\Lambda\subseteq{\mathbb{R}^{2d}}$ a lattice, and let
$\mathfrak{a}\in\mathscr{S}^{\prime}({\mathbb{R}^{2d}})$ be such that
$S_{g_{1},g_{2},\Lambda}=Op_{A}(\mathfrak{a})$. Then $\mathfrak{a}\in
M^{\infty,p_{0}}({\mathbb{R}^{2d}})$.

###### Proof.

In view of Remark 4.1 we may assume that $A=0$. The Weyl symbol of
$S_{g_{1},g_{2},\Lambda}$ with $g_{1}=g_{2}$ was calculated in [2, p.12], and
by similar arguments it follows that the Kohn-Nirenberg symbol is

$\displaystyle\mathfrak{a}(x,\xi)$
$\displaystyle=\sum_{(l,\lambda)\in\Lambda}g_{1}(x-l)\overline{\widehat{g}_{2}(\xi-\lambda)}e^{2\pi
i(x-l)\cdot(\lambda-\xi)}.$ (4.8)

In order to estimate the $M^{\infty,p_{0}}$ norm of $\mathfrak{a}$ we choose
the window $\Psi\in\mathcal{S}({\mathbb{R}^{2d}})\setminus\\{0\\}$ as

$\Psi(x,\xi):=\psi(x)\overline{\widehat{\psi}(\xi)}e^{-2\pi ix\cdot\xi}.$

By straightforward applications of Fourier inversion formula, it follows that

$\displaystyle(V_{\Psi}\mathfrak{a})(z,w)=\sum_{l,\lambda\in\Lambda}e^{-2\pi
i(y\cdot\xi+l\cdot\eta)}(V_{\psi}g_{1})(x-l,\xi+\eta-\lambda)\cdot\overline{(V_{\psi}g_{2})(x+y-l,\xi-\lambda)},$
(4.9)

(see e.g. [26]) with $z=(x,\xi)\in\mathbb{R}^{d}\times\mathbb{R}^{d}$ and
$w=(\eta,y)\in\mathbb{R}^{d}\times\mathbb{R}^{d}$.

Now let $\Lambda^{2}=\Lambda\times\Lambda$, and
$\Lambda^{4}=\Lambda^{2}\times\Lambda^{2}$ and choose
$\varepsilon=\frac{1}{N}$ with integer $N\geq 1$ large enough, such that

$\\{\psi(x-m)e^{2\pi
ix\cdot\mu}\\}_{m,\mu\in\varepsilon\Lambda}\quad\text{and}\quad\\{\Psi(z-w_{1})e^{2\pi
iz\cdot w_{2}}\\}_{w_{1},w_{2}\in\varepsilon\Lambda^{2}},$

are Gabor frames of $L^{2}(\mathbb{R}^{d})$ and $L^{2}({\mathbb{R}^{2d}})$,
respectively. Then

$\|g_{j}\|_{M^{p_{0}}}\asymp\|V_{\psi}g_{j}\|_{\ell^{p_{0}}(\varepsilon\Lambda^{2})}\quad\text{and}\quad\|\mathfrak{a}\|_{M^{\infty,p_{0}}}\asymp\|V_{\Psi}\mathfrak{a}\|_{\ell^{\infty,p_{0}}(\varepsilon\Lambda^{4})},$

in view of a result in [17].

Let $F_{j}=|V_{\psi}g_{j}|$. If $w_{1}=(m,\mu)\in\varepsilon\Lambda^{2}$ and
$w_{2}=(\nu,n)\in\varepsilon\Lambda^{2}$, then (4.9) yields

$\displaystyle|(V_{\Psi}\mathfrak{a})(w_{1},w_{2})|^{p_{0}}$
$\displaystyle\leq\sum_{l,\lambda\in\Lambda}F_{1}(m-l,\mu+\nu-\lambda)^{p_{0}}F_{2}(m+n-l,\mu-\lambda)^{p_{0}}$
$\displaystyle=(|\check{F}_{1}|^{p_{0}}*|F_{2}|^{p_{0}})(Tw_{2}),\qquad
Tw_{2}=(n,-\nu).$

The right-hand side does not depend on $w_{1}$, and therefore

$\|\mathfrak{a}\|_{M^{\infty,p_{0}}}\asymp\|V_{\Psi}\mathfrak{a}\|_{\ell^{\infty,p_{0}}(\varepsilon\Lambda^{4})}\lesssim\|{\check{F}}_{1}^{p_{0}}*F_{2}^{p_{0}}\|_{\ell^{1}(\varepsilon\Lambda^{2})}^{\frac{1}{p_{0}}}\\\\[4.30554pt]
\leq\big{(}\|F_{1}^{p_{0}}\|_{\ell^{1}(\varepsilon\Lambda^{2})}\|F_{2}^{p_{0}}\|_{\ell^{1}(\varepsilon\Lambda^{2}}\big{)}^{\frac{1}{p_{0}}}=\|F_{1}\|_{\ell^{p_{0}}(\varepsilon\Lambda^{2})}\|F_{2}\|_{\ell^{p_{0}}(\varepsilon\Lambda^{2})}\asymp\|g_{1}\|_{M^{p_{0}}}\|g_{2}\|_{M^{p_{0}}}\,.$

Thus $\mathfrak{a}\in M^{\infty,p_{0}}$.

###  4.4. Almost Diagonalization of Pseudodifferential Operators

In this section we list several characterizations of symbols in $\mathfrak{a}$
of $M^{\infty,p_{0}}({\mathbb{R}^{2d}})$, $p_{0}\in(0,\infty]$. The following
characterization of a symbol class by means of the almost diagonalization of
the associated pseudodifferential operator was found in [19, Theorem 3.2] for
$p_{0}=1$ and subsequently generalized to the full range of $p_{0}$ in [5,
Thm. 3.2] (with almost the same proof). This characterization helps to deduce
spectral properties of pseudodifferential operators from spectral properties
of infinite matrices. We only formulate the unweighted case of [5, Thm. 3.2],
which is sufficient for this paper.

######  Theorem 4.5.

(Almost Diagonalization). We fix a non-zero
$g\in\mathcal{S}({\mathbb{R}^{d}})\backslash\\{0\\}$ and a lattice
$\Lambda\subseteq{\mathbb{R}^{2d}}$ such that $\mathcal{G}(g,\Lambda)$ is a
Gabor frame for $L^{2}(\mathbb{R}^{d})$. Then for any $p_{0}\in(0,\infty]$ the
following properties are equivalent:

  * i)

$\mathfrak{a}\in M^{\infty,p_{0}}({\mathbb{R}^{2d}})$.

  * ii)

$\mathfrak{a}\in\mathcal{S}^{\prime}({\mathbb{R}^{2d}})$ and there exists a
function $H\in L^{p_{0}}({\mathbb{R}^{2d}})$, in fact $H\in
W(C_{b},L^{p_{0}})$, such that

$\displaystyle|\langle\mathfrak{a}^{w}\pi(z)g,\pi(w)g\rangle|\leq
H(w-z)\qquad\forall w,z\in{\mathbb{R}^{2d}}.$ (4.10)

  * iii)

$\mathfrak{a}\in\mathcal{S}^{\prime}({\mathbb{R}^{2d}})$ and there exists a
sequence $h\in\ell^{p_{0}}(\Lambda)$ such that

$\displaystyle|\langle\mathfrak{a}^{w}\pi(\rho)g,\pi(\lambda)g\rangle|\leq
h(\lambda-\rho)\qquad\forall\lambda,\rho\in\Lambda.$ (4.11)

As an immediate application we obtain the following result.

######  Corollary 4.6.

Under the hypotheses of Theorem 4.5 we assume that
$T:\mathcal{S}({\mathbb{R}^{d}})\rightarrow\mathcal{S}^{\prime}(\mathbb{R}^{d})$
is continuous and satisfies

$\displaystyle|\langle T\pi(\rho)g,\pi(\lambda)g\rangle|\leq
h(\lambda-\rho)\qquad\forall\lambda,\rho\in{\mathbb{Z}^{2d}}.$

for some $h\in\ell^{p_{0}}({\mathbb{Z}^{2d}})$. Then $T=\mathfrak{a}^{w}$ for
some $\mathfrak{a}\in M^{\infty,p_{0}}({\mathbb{R}^{2d}})$.

###### Proof.

Because of the continuity of
$T:\mathcal{S}({\mathbb{R}^{d}})\rightarrow\mathcal{S}^{\prime}(\mathbb{R}^{d})$
the Schwartz’s kernel theorem asserts the existence of a symbol
$\mathfrak{a}\in\mathcal{S}^{\prime}(\mathbb{R}^{2d})$ with
$T=\mathfrak{a}^{w}$. An application of Theorem 4.5 yields the claim.

###  4.5. Matrix Formulation

We fix a lattice $\Lambda$ and a window
$g\in\mathcal{S}({\mathbb{R}^{d}})\backslash\\{0\\}$ such that
$\mathcal{G}(g,\Lambda)$ is a frame with dual window $\gamma$ and associated
Gabor expansion (4.3) and (4.4).

For the manipulations to be meaningful, we assume of a symbol
$\mathfrak{a}\in\mathcal{S}^{\prime}(\mathbb{R}^{d})$ that the Weyl operator
$\mathfrak{a}^{w}$ is bounded on $M^{p}(\mathbb{R}^{d})$ for some
$p\in(0,\infty]$. Just keep in mind, that due to Proposition 4.1
$M^{p}(\mathbb{R}^{d})\subseteq M^{\infty}(\mathbb{R}^{d})$. On account of the
Gabor expansion (4.3) we have $D_{g}C_{\gamma}f=f$ for all $f\in
M^{p}(\mathbb{R}^{d})\subseteq M^{\infty}(\mathbb{R}^{d})$. Together with the
continuity of $\mathfrak{a}^{w}$ on $M^{p}(\mathbb{R}^{d})$ we obtain for all
$f\in M^{p}(\mathbb{R}^{d})$

$\displaystyle
C_{g}(\mathfrak{a}^{w}f)(\lambda)=\langle\mathfrak{a}^{w}f,\pi(\lambda)g\rangle=\sum_{\mu\in\Lambda}\langle
f,\pi(\mu)\gamma\rangle\langle\mathfrak{a}^{w}\pi(\mu)g,\pi(\lambda)g\rangle\quad\text{for
all }\lambda\in\Lambda.$ (4.12)

We define the matrix $M(\mathfrak{a})$ of $\mathfrak{a}^{w}$ with respect to
the frame $\mathcal{G}(g,\Lambda)$ by its entries

$\displaystyle
M(\mathfrak{a})_{\lambda,\mu}=\langle\mathfrak{a}^{w}\pi(\mu)g,\pi(\lambda)g\rangle\qquad\text{for
all }\lambda,\mu\in\Lambda\,.$

We can then recast (4.12) as

$\displaystyle C_{g}(\mathfrak{a}^{w}f)=M(\mathfrak{a})C_{\gamma}f.$ (4.13)

Likewise, Theorem 4.5 now reads as follows.

######  Corollary 4.7.

Let $\mathfrak{a}\in\mathcal{S}^{\prime}({\mathbb{R}^{2d}})$. Then
$\mathfrak{a}\in M^{\infty,p_{0}}({\mathbb{R}^{2d}})$, if and only if
$M(\mathfrak{a})\in\mathcal{C}^{p_{0}}$.

The matrix $P$ associated to the identity operator has the entries

$\displaystyle
P_{\lambda,\mu}=\langle\pi(\mu)g,\pi(\lambda)g\rangle,\qquad\text{for all
}\lambda,\mu\in\Lambda.$ (4.14)

To simplify the analysis of $P$, we assume from now on that
$\mathcal{G}(g,\Lambda)$ is a tight frame with $S_{g,\Lambda}=\mathrm{I}$. As
mentioned already, tight frames with a window in $\mathcal{S}(\mathbb{R}^{d})$
always exist. The matrix $P$ has the following properties.

######  Lemma 4.8.

Assume that $\mathcal{G}(g,\Lambda)$ is a tight frame with
$g\in\mathcal{S}(\mathbb{R}^{d})$. Let $0<p_{0}\leq p\leq\infty$ and
$\mathfrak{a}\in\mathcal{S}^{\prime}({\mathbb{R}^{2d}})$ be such that the
associated pseudodifferential operator $\mathfrak{a}^{w}$ is bounded on
$M^{p}$. Then

  * (i)

$P$ is a projection from $\ell^{p}(\Lambda)$ to range $C_{g}(M^{p})$, i.e.,
$Pc=c\in\ell^{p}(\Lambda)$ if and only if there exists $f\in M^{p}$ such that
$c_{\lambda}=\langle f,\pi(\lambda)g\rangle=(C_{g}f)_{\lambda}$ for
$\lambda\in\Lambda$.

  * (ii)

$PM(\mathfrak{a})=M(\mathfrak{a})$ and $M(\mathfrak{a})P=M(\mathfrak{a})$.

  * (iii)

$P\in\mathcal{C}^{p}(\Lambda)$.

  * (iv)

For $\mathfrak{a}\in M^{\infty,p_{0}}({\mathbb{R}^{2d}})$ we have
$M(\mathfrak{a})+I-P\in\mathcal{C}^{p_{0}}(\Lambda)$.

###### Proof.

(i) For all $\lambda,\nu\in\Lambda$ the assumption that
$S_{g,\Lambda}=\mathrm{Id}$ and (4.2) imply that

$\displaystyle(P^{2})_{\lambda,\nu}$
$\displaystyle=\Big{\langle}\sum_{\mu\in\Lambda}\langle\pi(\nu)g,\pi(\mu)g\rangle\pi(\mu)g,\pi(\lambda)g\Big{\rangle}=\langle
S_{g,\Lambda}\pi(\nu)g,\pi(\lambda)g\rangle=\langle\pi(\nu)g,\pi(\lambda)g\rangle$
$\displaystyle=P_{\lambda,\nu}.$

Consequently $P^{2}=P$ and $P$ is a projection. Next let
$c\in\ell^{p}(\Lambda)$ with $Pc=c$. With
$f=\sum_{\mu\in\Lambda}c_{\mu}\pi(\mu)g$ we have for $\lambda\in\Lambda$,

$\displaystyle
c_{\lambda}=(Pc)_{\lambda}=\sum_{\mu\in\Lambda}\langle\pi(\mu)g,\pi(\lambda)g\rangle
c_{\mu}=\langle f,\pi(\lambda)g\rangle=(C_{g}f)_{\lambda}\,,$

whence $c=C_{g}f$. Then the norm-equivalence (4.5) yields $f\in
M^{p}(\mathbb{R}^{d})$.

Conversely, assume that $c=C_{g}f$ for some $f\in M^{p}(\mathbb{R}^{d})$. Then
we get due to (4.3)

$\displaystyle(Pc)_{\lambda}=\langle\sum_{\mu\in\Lambda}\langle
f,\pi(\mu)g\rangle\pi(\mu)g,\pi(\lambda)g\rangle=\langle
S_{g,\Lambda}f,\pi(\lambda)g\rangle=\langle
f,\pi(\lambda)g\rangle=c_{\lambda},\quad\lambda\in\Lambda\,,$

and (i) holds.

(ii) is proved similarly by straight forward calculations using (4.2).

(iii) If $g\in\mathcal{S}(\mathbb{R}^{d})$, then also
$V_{g}g\in\mathcal{S}({\mathbb{R}^{2d}})$, e.g., by [18, Thm. 11.2.5]. In
particular, this implies that for every $N\geq 0$

$|V_{g}g(z)|\lesssim(1+|z|)^{-N}\,,$

and thus

$|P_{\lambda,\mu}|=|\langle\pi(\mu)g,\pi(\lambda)g\rangle|=|V_{g}g(\lambda-\mu)|\lesssim(1+|\lambda-\mu|)^{-N}\,.$

By choosing $N$ large enough, we see that $H(z)=(1+|z|)^{-N}$ is in
$W(C_{b},L^{p_{0}})$ and consequently $P\in\mathcal{C}^{p_{0}}(\Lambda)$.

(iv) Since all matrices $P,\mathrm{I}$ and $M(\mathfrak{a})$ are in
$\mathcal{C}^{p_{0}}$, their sum $M(\mathfrak{a})+\mathrm{Id}+P$ is also in
$\mathcal{C}^{p_{0}}$.

###  4.6. Spectral invariance.

We have already seen before that it is possible to relate a pseudodifferential
operator $\mathfrak{a}^{w}$ to an infinite matrix $M(\mathfrak{a})$. It turns
out that there is a connection between the invertibility of $\mathfrak{a}^{w}$
and $M(\mathfrak{a})$.

######  Lemma 4.9.

Assume that $\mathcal{G}(g,\Lambda)$ is a tight frame with
$g\in\mathcal{S}(\mathbb{R}^{d})$. Let $0<p\leq\infty$ and
$\mathfrak{a}\in\mathcal{S}^{\prime}({\mathbb{R}^{2d}})$ be such that the
associated Weyl operator $\mathfrak{a}^{w}$ is bounded on $M^{p}$. Then
$\mathfrak{a}^{w}$ is invertible on $M^{p}$, if and only if

  * i)

$\|M(\mathfrak{a})Pc\|_{p}\gtrsim\|Pc\|_{p}\quad\text{ for all
}c\in\ell^{p}(\Lambda)$ and

  * ii)

for every $c_{0}\in P\ell^{p}(\Lambda)$, there is a $c\in P\ell^{p}(\Lambda)$
such that $M(\mathfrak{a})Pc=Pc_{0}$.

Here the projection $P$ is defined as in (4.14).

###### Proof.

Let $\mathfrak{a}^{w}$ be invertible on $M^{p}(\mathbb{R}^{d})$. Using (4.13),
we obtain

$\displaystyle\|M(\mathfrak{a})C_{g}f\|_{p}=\|C_{g}(\mathfrak{a}^{w}f)\|_{p}\asymp\|\mathfrak{a}^{w}f\|_{M^{p}}\gtrsim\|f\|_{M^{p}}\asymp\|C_{g}f\|_{p}\,,$

which is (i). To prove (ii), let $c_{0}\in P\ell^{p}(\Lambda)$ be arbitrary.
Then there exists $h\in M^{p}(\mathbb{R}^{d})$ with $C_{g}h=c_{0}=Pc_{0}$ by
Lemma 4.8. Since $\mathfrak{a}^{w}$ is bijective on $M^{p}(\mathbb{R}^{d})$,
there is a $f\in M^{p}(\mathbb{R}^{d})$ such that

$\displaystyle\mathfrak{a}^{w}f=h.$

Then we obtain for $c=C_{g}f$ due to Lemma 4.8 that $Pc=c$ and that

$\displaystyle
M(\mathfrak{a})Pc=M(\mathfrak{a})c=M(\mathfrak{a})C_{g}f=C_{g}(\mathfrak{a}^{w}f)=C_{g}h=c_{0}=Pc_{0}.$

This implies (ii).

Conversely assume that (i) and (ii) hold. Using (4.13) (with $\gamma=g$) and
(i), we obtain for all $f\in M^{p}(\mathbb{R}^{d})$

$\displaystyle\|\mathfrak{a}^{w}f\|_{M^{p}}\asymp\|C_{g}(\mathfrak{a}^{w}f)\|_{p}=\|M(\mathfrak{a})C_{g}f\|_{p}\gtrsim\|C_{g}f\|_{p}\asymp\|f\|_{M^{p}}.$

Hence $\mathfrak{a}^{w}$ is one-to-one on $M^{p}(\mathbb{R}^{d})$. To prove
that $\mathfrak{a}^{w}$ is surjective we choose an arbitrary $h\in
M^{p}(\mathbb{R}^{d})$ and let $c_{0}:=C_{g}h\in\ell^{p}(\Lambda)$. Then
$c_{0}=Pc_{0}\in P\ell^{p}(\Lambda)$.

By assumption (ii) there is a $c\in P\ell^{p}(\Lambda)$ such that

$M(\mathfrak{a})Pc=Pc_{0}=c_{0}=C_{g}h.$ (4.15)

Since the image of $P$ is $C_{g}(M^{p})$, there is a $f\in
M^{p}(\mathbb{R}^{d})$ with

$\displaystyle Pc=C_{g}f.$

A combination of (4.3), (4.13) and (4.15) yields

$\displaystyle\mathfrak{a}^{w}f=D_{g}C_{g}(\mathfrak{a}^{w}f)=D_{g}M(\mathfrak{a})Pc=D_{g}C_{g}h=h.$

This implies that $\mathfrak{a}^{w}$ maps onto $M^{p}(\mathbb{R}^{d})$ and
hence invertible on $M^{p}(\mathbb{R}^{d})$.

In the previous lemma we proved the equivalence of the invertibility of
$\mathfrak{a}^{w}$ on $M^{p}(\mathbb{R}^{d})$ and of $M(\mathfrak{a})$ on
$P\ell^{p}(\Lambda)$. Since $\mathrm{ker}\,P\neq\\{0\\}$ and
$M(\mathfrak{a})=M(\mathfrak{a})P$, $M(\mathfrak{a})$ cannot be invertible on
the whole space $\ell^{p}(\Lambda)$. In the literature this problem is usually
overcome by using the pseudo-inverse of $M(\mathfrak{a})$ and holomorphic
functional calculus. Here we use a new trick, which may be of independent
interest. Consider the matrix $A=M(\mathfrak{a})+\mathrm{Id}-P$. We can then
use the spectral invariance result for infinite convolution-dominated matrices
of Theorem 3.15 to derive a spectral invariance result for pseudodifferential
operators on modulation spaces, cf. Theorem 1.2.

###### Proof of Theorem 1.2.

Let $p\in[p_{0},\infty]$ be the index for which $\mathfrak{a}^{w}$ is
invertible on $M^{p}(\mathbb{R}^{d})$ and let
$A=M(\mathfrak{a})+\mathrm{Id}-P$, where $P$ is the projection defined in
(4.14). First we check the assumptions of Theorem 3.15 and prove that

$\displaystyle A=M(\mathfrak{a})+I-P\qquad\text{ is invertible on
}\ell^{p}(\Lambda).$ (4.16)

Assume that $Ac=0$ for some $c\in\ell^{p}(\Lambda)$. Then by i) and ii) of
Lemma 4.8

$\displaystyle
0=Ac=\left(M(\mathfrak{a})+I-P\right)\left(Pc+(I-P)c\right)=M(\mathfrak{a})Pc+(I-P)c.$

If we apply $P$ respectively $I-P$ to the previous equality and use Lemma 4.8
again, we obtain

$\displaystyle M(\mathfrak{a})Pc=0\qquad\text{and}\qquad(I-P)c=0.$

Since $\mathfrak{a}^{w}$ is invertible on $M^{p}$ we obtain by Lemma 4.9 and
the previous estimate

$\displaystyle\|Pc\|_{p}\lesssim\|M(\mathfrak{a})Pc\|_{p}=0,$

and consequently $Pc=0$. Hence $c=Pc+(I-P)c=0$ which shows that $A$ is one-to-
one.

To show the surjectivity of $A$, we let $c_{0}\in\ell^{p}(\Lambda)$ be
arbitrary. Since $\mathfrak{a}^{w}$ is invertible on $M^{p}$, Lemma 4.9 yields
the existence of $c\in P\ell^{p}(\Lambda)$ with

$\displaystyle M(\mathfrak{a})Pc=Pc_{0}.$ (4.17)

Then $\widetilde{c}=Pc+(I-P)c_{0}\in\ell^{p}(\Lambda)$ and by Lemma 4.8 and
(4.17)

$\displaystyle
A\widetilde{c}=M(\mathfrak{a})\widetilde{c}+(I-P)\widetilde{c}=M(\mathfrak{a})Pc+(I-P)c_{0}=Pc_{0}+(I-P)c_{0}=c_{0}.$

Thus $A$ is onto on $\ell^{p}(\Lambda)$ and therefore invertible on
$\ell^{p}(\Lambda)$. Due to Lemma 4.8 we have
$A\in\mathcal{C}^{p_{0}}(\Lambda)$. Since (4.16) also holds we can apply
Theorem 3.15 and get the invertibility of $A$ on $\ell^{q}(\Lambda)$ for all
$q\in[p_{0},\infty)$.

Next we show, that

$\displaystyle M(\mathfrak{a})\text{ is invertible on
}P\ell^{q}(\Lambda)\text{ for all }q\in[p_{0},\infty).$ (4.18)

Let $q\in[p_{0},\infty)$ be arbitrary. Since $I-P\equiv 0$ on
$P\ell^{q}(\Lambda)$, the injectivity of $A$ implies that $M(\mathfrak{a})$ is
one-to-one on $P\ell^{q}(\Lambda)$. For an arbitrary $c_{0}\in
P\ell^{q}(\Lambda)\subseteq\ell^{q}(\Lambda)$, there is some
$c\in\ell^{q}(\Lambda)$ with $Ac=c_{0}$, since $A$ is onto on
$\ell^{q}(\Lambda)$. Since $(I-P)M(\mathfrak{a})=0$ by Lemma 4.8, we obtain,
after applying $I-P$ to $Ac=c_{0}$, that $0=(I-P)c_{0}=(I-P)Ac=(I-P)c$. Then
$c\in P\ell^{q}(\Lambda)$ and

$\displaystyle M(\mathfrak{a})c=M(\mathfrak{a})c+(I-P)c=Ac=c_{0}.$

Hence $M(\mathfrak{a})$ is onto $P\ell^{q}(\Lambda)$ and (4.18) holds. By
Lemma 4.9 $\mathfrak{a}$ is invertible on $M^{q}$.

We now prove Theorem 1.1 and obtain more refined information about the inverse
$(\mathfrak{a}^{w})^{-1}$.

###### Proof of Theorem 1.1.

By Theorem 1.2 we get the invertibility of $\mathfrak{a}^{w}$ on
$M^{2}(\mathbb{R}^{d})=L^{2}(\mathbb{R}^{d})$. By [19, Theorem 4.6] and the
embedding $M^{\infty,p_{0}}(\mathbb{R}^{2d})\subseteq
M^{\infty,1}(\mathbb{R}^{2d})$, there is a symbol $\mathfrak{b}\in
M^{\infty,1}({\mathbb{R}^{2d}})$ with
$\mathfrak{b}^{w}=(\mathfrak{a}^{w})^{-1}$. We consider the associated
matrices $M(\mathfrak{a})$ and $M(\mathfrak{b})$ with respect to a tight Gabor
frame $\mathcal{G}(g,\Lambda)$ with $g\in\mathcal{S}(\mathbb{R}^{d})$ and
again denote by $P$ the projection with entries
$P_{\lambda\mu}=\langle\pi(\mu)g,\pi(\lambda)g\rangle$. On account of Lemma
4.8 we get for all $c\in\text{ran }C_{g}$ the existence of an $f\in
M^{2}(\mathbb{R}^{d})$ with $c=C_{g}f=Pc$. Then for all
$c=C_{g}f=Pc\in\text{ran }C_{g}$ we obtain using (4.13)

$\displaystyle
M(\mathfrak{b})M(\mathfrak{a})c=M(\mathfrak{b})M(\mathfrak{a})C_{g}f=M(\mathfrak{b})C_{g}(\mathfrak{a}^{w}f)=C_{g}(\mathfrak{b}^{w}\mathfrak{a}^{w}f)=C_{g}f=c\,.$

If $Pc=0$, then $M(\mathfrak{a})c=M(\mathfrak{a})Pc=0$, consequently on
$\ell^{2}(\Lambda)$ we have

$M(\mathfrak{b})M(\mathfrak{a})=P\,.$

It follows that

$\big{(}M(\mathfrak{b})+\mathrm{Id}-P\big{)}\big{(}M(\mathfrak{a})+\mathrm{Id}-P\big{)}=\big{(}M(\mathfrak{b})+\mathrm{Id}-P\big{)}\,A=\mathrm{Id}\,.$

This means that $B=M(\mathfrak{b})+\mathrm{Id}-P$ is the inverse of the
invertible matrix $A$ (since the inverse is unique). Since
$A\in\mathcal{C}^{p_{0}}$ by Lemma 4.8, Theorem 3.14 implies that also
$B\in\mathcal{C}^{p_{0}}$.

Consequently $M(\mathfrak{b})\in\mathcal{C}^{p_{0}}$. Now the characterization
of Corollary 4.7 implies that $\mathfrak{b}\in
M^{\infty,p_{0}}({\mathbb{R}^{2d}})$, as claimed.

By using Theorem 1.1 we can now deduce the invertibility of $\mathfrak{a}$ on
more general modulation spaces which generalizes Theorem 1.2.

######  Theorem 4.10 (Spectral invariance on Modulation Spaces).

If $\mathfrak{a}\in M^{\infty,p_{0}}({\mathbb{R}^{2d}})$ for $p_{0}\in(0,1]$
and $\mathfrak{a}^{w}$ is invertible on $M^{p}(\mathbb{R}^{d})$ for some
$p\in[p_{0},\infty]$, then $\mathfrak{a}^{w}$ is also invertible on
$M^{r,q}(\mathbb{R}^{d})$ for all $r,q\in[p_{0},\infty)$.

###### Proof.

On account of Theorem 1.1 there is a $\mathfrak{b}\in
M^{\infty,p_{0}}({\mathbb{R}^{2d}})$ with
$\mathfrak{b}^{w}=(\mathfrak{a}^{w})^{-1}$ on $M^{p}(\mathbb{R}^{d})$. By
Proposition 4.3 $\mathfrak{b}^{w}$ is bounded on $M^{p,q}(\mathbb{R}^{d})$.
Since $\mathfrak{b}^{w}\mathfrak{a}^{w}=\mathfrak{a}^{w}\mathfrak{b}^{w}=I$ on
$\mathcal{S}({\mathbb{R}^{d}})\subseteq M^{p}(\mathbb{R}^{d})$ we obtain the
invertibility of $\mathfrak{a}^{w}$ on $M^{p,q}(\mathbb{R}^{d})$ by the
density of $\mathcal{S}({\mathbb{R}^{d}})$ in $M^{p,q}(\mathbb{R}^{d})$.

This theorem is an extension of [19, Corollary 4.7] from the case $p_{0}=1$ to
$p_{0}<1$. By using the arguments of [24], one can formulate the corollary for
an even more general class of modulation spaces.

######  Remark 4.2.

Let $A\in\mathbb{R}^{d\times d}$. Proposition 4.2 implies that the conclusions
in Theorems 1.1, 1.2 and Theorem 4.10 remain true with $Op_{A}(\mathfrak{a})$
and $Op_{A}(\mathfrak{b})$ in place of $\mathfrak{a}^{w}$ and
$\mathfrak{b}^{w}$, respectively, at each occurrence.

As an application of Theorem 4.10 we show the following property of the
canonical dual window of an Gabor frame.

######  Theorem 4.11.

Let $0<p\leq 1$, $\Lambda\subseteq{\mathbb{R}^{2d}}$ be a lattice and $g\in
M^{p}(\mathbb{R}^{d})$ be such that $\mathcal{G}(g,\Lambda)$ is a Gabor frame
for $L^{2}(\mathbb{R}^{d})$. Then the canonical dual window

$\displaystyle\gamma=S_{g,\Lambda}^{-1}g\in M^{p}(\mathbb{R}^{d}).$ (4.19)

###### Proof.

We denote the Kohn-Nirenberg symbol of the frame operator $S_{g,\Lambda}$ by
$\mathfrak{a}$. By Proposition 4.4 we have

$\displaystyle\mathfrak{a}\in M^{\infty,p}({\mathbb{R}^{2d}}).$

Since Theorem 4.10 also holds for pseudodifferential operators in the Kohn-
Nirenberg quantization, $S_{g,\Lambda}$ is invertible on
$M^{p}(\mathbb{R}^{d})$. Therefore (4.19) holds.

##  Appendix A Proofs of some preparatory results

In this appendix we prove some preparatory results from Section 2 and 3.

###### Proof of Lemma 2.2.

Let $\alpha>0$ be chosen such that $[0,\alpha]^{d}\subseteq B_{1/2}(0)$ and
$(B_{1/2}(\alpha k))_{k\in\mathbb{Z}^{d}}$ covers $\mathbb{R}^{d}$. Then

$\displaystyle\|H\|_{\ell^{p_{0}}(\Lambda)}^{p_{0}}\leq\sum_{k\in\mathbb{Z}^{d}}\sum_{\Lambda\cap
B_{1/2}(\alpha
k)}|H(\lambda)|^{p_{0}}\lesssim\sum_{k\in\mathbb{Z}^{d}}\mathop{\mathrm{rel}}(\Lambda)\int_{[0,\alpha]^{d}}\sup_{\lambda\in
B_{1/2}(\alpha k)}|H(\lambda)|^{p_{0}}dy.$

Since $B_{1/2}(\alpha k)\subseteq B_{1}(\alpha k+y)$ for each
$y\in[0,\alpha]^{d}$ the integral just becomes larger, if we take the supremum
over all $\lambda\in B_{1}(\alpha k+y)$ instead of $\lambda\in B_{1/2}(\alpha
k)$. A substitution then provides

$\displaystyle\|H\|_{\ell^{p_{0}}(\Lambda)}^{p_{0}}\lesssim\sum_{k\in\mathbb{Z}^{d}}\mathop{\mathrm{rel}}(\Lambda)\int_{[0,\alpha]^{d}+\alpha
k}\sup_{\lambda\in
B_{1}(x)}|H(\lambda)|^{p_{0}}\,dx=\mathop{\mathrm{rel}}(\Lambda)\|H\|_{W(C_{b},L^{p_{0}})}^{p_{0}}.$

###### Proof of Proposition 3.4.

For $\rho_{0}\in\Pi$ and $\lambda_{0}\in\Lambda$ fixed let

$s_{\Lambda,p_{0}}(\rho_{0})^{p_{0}}:=\sum_{\lambda\in\Lambda}H(\lambda-\rho_{0})^{p_{0}}\quad\text{and}\quad
s_{\Pi,p_{0}}(\lambda_{0})^{p_{0}}:=\sum_{\rho\in\Pi}H(\lambda_{0}-\rho)^{p_{0}}.$

Then it follows by straightforward estimates that

$s_{\Lambda,p_{0}}(\rho)\leq C_{\Lambda}\|H\|_{W(C_{b},L^{p_{0}})},\quad
s_{\Pi,p_{0}}(\lambda)\leq
C_{\Pi}\|H\|_{W(C_{b},L^{p_{0}})},\quad\lambda\in\Lambda,\ \rho\in\Pi,$

where $C_{\Lambda}=C_{0}\operatorname{rel}(\Lambda)^{\frac{1}{p_{0}}}$, for
some constant $C_{0}>0$ which only depends on $d$.

Suppose $q\geq 1$. Then Hölder’s inequality together with the facts that
$s_{\Lambda,p_{0}}$ and $s_{\Pi,p_{0}}$ decrease with $p_{0}$ give

$\displaystyle\|Ab\|_{\ell^{q}(\Lambda)}^{q}$
$\displaystyle\leq\sum_{\lambda\in\Lambda}\left(\sum_{\rho\in\Pi}(H(\lambda-\rho)^{\frac{1}{q}}|b_{\rho}|)H(\lambda-\rho)^{\frac{1}{q^{\prime}}}\right)^{q}$
$\displaystyle\leq\sum_{\lambda\in\Lambda}\left(\sum_{\rho\in\Pi}H(\lambda-\rho)|b_{\rho}|^{q}\right)\left(\sum_{\rho\in\Pi}H(\lambda-\rho)\right)^{\frac{q}{q^{\prime}}}=\sum_{\lambda\in\Lambda}\left(\sum_{\rho\in\Pi}H(\lambda-\rho)|b_{\rho}|^{q}\right)s_{\Pi,1}(\lambda)^{\frac{q}{q^{\prime}}}$
$\displaystyle\leq(C_{\Pi}\|H\|_{W(C_{b},L^{p_{0}})})^{\frac{q}{q^{\prime}}}\sum_{\rho\in\Pi}\left(\sum_{\lambda\in\Lambda}H(\lambda-\rho)\right)|b_{\rho}|^{q}$
$\displaystyle=(C_{\Pi}\|H\|_{W(C_{b},L^{p_{0}})})^{\frac{q}{q^{\prime}}}\sum_{\rho\in\Pi}s_{\Lambda,1}(\rho)|b_{\rho}|^{q}\leq
C_{\Pi}^{\frac{q}{q^{\prime}}}\|H\|_{W(C_{b},L^{p_{0}})}^{q}C_{\Lambda}\|b\|_{\ell^{q}(\Pi)}^{q},$

giving the assertion when $q\geq 1$.

If instead $p_{0}\leq q\leq 1$, then

$\displaystyle\|Ab\|_{\ell^{q}(\Lambda)}^{q}$
$\displaystyle\leq\sum_{\lambda\in\Lambda}\left(\sum_{\rho\in\Pi}H(\lambda-\rho)|b_{\rho}|\right)^{q}\leq\sum_{\rho\in\Pi}\left(\sum_{\lambda\in\Lambda}H(\lambda-\rho)^{q}\right)|b_{\rho}|^{q}$
$\displaystyle\leq\sum_{\rho\in\Pi}s_{\Lambda,q}(\rho)^{q}|b_{\rho}|^{q}\leq
C_{\Lambda}^{q}\|H\|_{W(C_{b},L^{p_{0}})}^{q}\sum_{\rho\in\Pi}|b_{\rho}|^{q}=(C_{\Lambda}\|H\|_{W(C_{b},L^{p_{0}})}\|b\|_{\ell^{q}(\Pi)})^{q},$

giving the result for $q\leq 1$.

###### Proof of Lemma 3.6 for $q<1$.

Since $\operatorname{supp}(\varphi)\subseteq B_{2}(0)$, we get

$\displaystyle\eta:=\sup_{\varepsilon>0}\sup_{x\in\mathbb{R}^{d}}\\#\big{\\{}\,k\in\mathbb{Z}^{d}\,:\,\varphi^{\varepsilon}_{k}(x)\not=0\,\big{\\}}=\sup_{\varepsilon>0}\sup_{x\in\mathbb{R}^{d}}\\#\\{k\in\mathbb{Z}^{d}\cap
B_{2}(\varepsilon x)\\}<\infty.$ (A.1)

So we obtain the following bound for all $x\in\mathbb{R}^{d}$:

$\displaystyle
1\leq\sum_{k\in\mathbb{Z}^{d}}\varphi^{\varepsilon}_{k}(x)^{q}=\sum_{k\in\mathbb{Z}^{d}:\varphi^{\varepsilon}_{k}(x)\not=0}\varphi^{\varepsilon}_{k}(x)^{q}\leq\eta\,\sup_{k\in\mathbb{Z}^{d}}\varphi^{\varepsilon}_{k}(x)^{q}\leq\eta.$

Therefore, for all $x\in\mathbb{R}^{d}$,

$\displaystyle\frac{1}{\eta}\leq\sup_{k\in\mathbb{Z}^{d}}\varphi^{\varepsilon}_{k}(x)^{q}\leq\sum_{k\in\mathbb{Z}^{d}}\varphi^{\varepsilon}_{k}(x)^{q}\leq\eta\,.$
(A.2)

If $a\in\ell^{q}(\Pi)$, then

$\displaystyle\frac{1}{\eta}\sum_{\rho\in\Pi}\lvert
a_{\rho}\rvert^{q}\leq\sum_{\rho\in\Pi}\sum_{k\in\mathbb{Z}^{d}}\varphi^{\varepsilon}_{k}(\rho)^{q}\lvert
a_{\rho}\rvert^{q}\leq\eta\sum_{\rho\in\Pi}\lvert a_{\rho}\rvert^{q},$

which implies the claim with constants independent of $\varepsilon$ and $q$.

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