Quasi-Banach algebras and Wiener properties for pseudodifferential and generalized metaplectic operators

Abstract

We generalize the results for Banach algebras of pseudodifferential operators obtained by Gröchenig and Rzeszotnik (Ann Inst Fourier 58:2279–2314, 2008) to quasi-algebras of Fourier integral operators. Namely, we introduce quasi-Banach algebras of symbol classes for Fourier integral operators that we call generalized metaplectic operators, including pseudodifferential operators. This terminology stems from the pioneering work on Wiener algebras of Fourier integral operators (Cordero et al. in J Math Pures Appl 99:219–233, 2013), which we generalize to our framework. This theory finds applications in the study of evolution equations such as the Cauchy problem for the Schrödinger equation with bounded perturbations, cf. (Cordero, Giacchi and Rodino in Wigner analysis of operators. Part II: Schrödinger equations, arXiv:2208.00505).

Appendix A. Quasi-Banach algebras

We consider here the solid involutive quasi-Banach algebras with respect to convolution $\mathcal {B}=\ell ^q_{v_{s}}(\Lambda )$, $s\ge 0$, $0<q\le 1$. For $q=1$ we recapture the algebra $\ell ^1_{v_s}(\Lambda )$. As before, without loss of generality, we may assume $\Lambda ={\mathbb {Z}^{2d}}$.

The unit element is given by the sequence $\delta =(\delta (k))_{k\in {\mathbb {Z}^{2d}}}$, with elements

$$\begin{aligned} \delta (k)= {\left\{ \begin{array}{ll} 1,\quad k=0\\ 0,\quad k\in {\mathbb {Z}^{2d}}{\setminus }\{0\}, \end{array}\right. } \end{aligned}$$

We have $\Vert \delta ||_{\ell ^q_{v_s}}=1$ for every $s\in \mathbb {R}$. Moreover, for every $a\in \ell ^q_{v_s}({\mathbb {Z}^{2d}})$,

$$\begin{aligned} a*\delta (k)=\sum _{j\in {\mathbb {Z}^{2d}}} \delta (j)a(k-j) = \delta (0)a(k) =a(k), \end{aligned}$$

$k\in {\mathbb {Z}^{2d}}$.

For sake of clarity, we first recall the general definition of a quasi-Banach space.

Definition A.1

Let be X a complex vector space. A functional $\Vert \cdot \Vert : X\rightarrow [0,+\infty )$ is called quasinorm if the following inequality holds

$$\begin{aligned} \Vert f+g\Vert \le K (\Vert f\Vert +\Vert g\Vert ),\quad \forall f,g\in X, \end{aligned}$$

(39)

where $K\ge 1$, moreover,

$$\begin{aligned} \Vert f\Vert \ge 0\quad \text{ and } \quad \Vert f\Vert =0\Leftrightarrow f=0 \end{aligned}$$

and

$$\begin{aligned} \Vert \lambda f\Vert =|\lambda |\Vert f\Vert ,\quad \forall \lambda \in \mathbb {C},\,f\in X. \end{aligned}$$

The couple $(x,\Vert \cdot \Vert )$ is called a quasinormed space. A complete quasinormed vector space is called a quasi-Banach space.

Standard examples are $L^q$ spaces with $0<q<1$. In this case the functional $\Vert \cdot \Vert =\Vert \cdot \Vert _{L^q}$ is not a norm but satisfies (39) with $K=2^{1/q}-1>1$ and it holds

$$\begin{aligned} \Vert f+g\Vert ^q\le \Vert f\Vert ^q+\Vert g\Vert ^q,\quad \forall f,g\in L^q. \end{aligned}$$

(40)

A functional satisfying (39) and (40) is called a q-norm. Relation (40) generalizes to

$$\begin{aligned} \Vert f_1+f_2+\cdots f_n\Vert ^q\le \sum \limits _{1}^{n} \Vert f_n\Vert ^q,\quad \forall f_i\in L^q,\, i=1,\dots ,n. \end{aligned}$$

(41)

If the metric $d(f,g)=|||f-g|||^q$ on X defines a metric that induces the same topology on the quasi-Banach space $(X,\Vert \cdot \Vert )$, then X is also called q-Banach space.

Theorem A.2

(Aoki-Rolewicz [29]) If $\Vert \cdot \Vert$ is a quasinorm on X, then there exist $q>0$ and a q-norm $|||\cdot |||$ on X such that

$$\begin{aligned} \frac{1}{C}\Vert f\Vert \le |||f|||\le \Vert f\Vert ,\quad f\in X, \end{aligned}$$

where $C>0$ is independent of f.

Following the pattern of [25], from now on we assume that quasinorm means q-norm, for some $q\in (0,1]$.

1.1 A.1. General theory of quasi-Banach algebras

Definition A.3

A (complex) quasi-Banach algebra $\mathcal {A}$ is a complex vector space in which a multiplication $\cdot :\mathcal {A}\times \mathcal {A}\rightarrow \mathcal {A}$ is satisfied so that

(i) $x\cdot (y\cdot z)=(x\cdot y)\cdot z$,

(ii) $(x+y)\cdot z=x\cdot z+y\cdot z$,

(iii) $\alpha (x\cdot y)=(\alpha x)\cdot y=x\cdot (\alpha y)$ for all $x,y,z\in \mathcal {A}$ and $\alpha \in \mathbb {C}$. In addition, $\mathcal {A}$ is a quasi-Banach space with respect to a quasi-norm $\Vert \cdot \Vert$ that satisfies

$$\begin{aligned} \Vert x\cdot y\Vert \le C_P\Vert x\Vert \Vert y\Vert \end{aligned}$$

(42)

for some $C_P>0$, and $\mathcal {A}$ contains an element e such that

(iv) $x\cdot e=e\cdot x=x$; (v) $\Vert e\Vert =1$.

For $C_P=1$ (42) becomes $\Vert x\cdot y\Vert \le \Vert x\Vert \Vert y\Vert$ and we have the standard algebra property. In particular, if $C_P\le 1$ then the estimate $\Vert x\cdot y\Vert \le \Vert x\Vert \Vert y\Vert$ holds as well. Thus, we limit to the case

$$\begin{aligned} C_P\ge 1. \end{aligned}$$

In what follows, $\mathcal {A}$ will always denote a quasi-Banach agebra and $C_P$ will always denote the constant that appears in (42). Also, we denote with $C_S$ the constant in the definition of quasi-norm, namely

$$\begin{aligned} \Vert x+y\Vert \le C_S\Vert x\Vert \Vert y\Vert . \end{aligned}$$

Examples for the case $C_S=1$ is given by $A=\ell ^q_{v_s}({\mathbb {Z}^{2d}})$, $0<q\le 1$, $s\ge 0$, which satisfies:

$$\begin{aligned} \Vert x*y\Vert _{\ell ^q_{v_s}}\le \Vert x\Vert _{\ell ^q_{v_s}} \Vert y\Vert _{\ell ^q_{v_s}}. \end{aligned}$$

Moreover $\ell ^q_{v_s}({\mathbb {Z}^{2d}})$ are q-Banach spaces.

From now on, we may assume without loss of generality that $\Lambda ={\mathbb {Z}^{2d}}$.

Remark A.4

The multiplication $\cdot :\mathcal {A}\times \mathcal {A}\rightarrow \mathcal {A}$ is continuous with respect to the quasi-norm topology on $\mathcal {A}$ and left/right continuous. The proof goes exactly as in the Banach case.

[27, Proposition 10.6] extends to the quasi-Banach case directly. For the following theorem in the Banach setting, we refer to [27, Theorem 10.7].

Recall that a complex homomorphism on a quasi-Banach algebra $\mathcal {A}$ is a linear mapping $\phi :\mathcal {A}\rightarrow \mathbb {C}$ such that $\phi \not \equiv 0$ and $\phi (x\cdot y)=\phi (x)\phi (y)$ for all $x,y\in \mathcal {A}$.

Theorem A.5

Let $\mathcal {A}$ be a quasi-Banach algebra, $x\in \mathcal {A}$, $\Vert x\Vert <\frac{1}{C_P}$. Then,

(i) $e-x$ is invertible in $\mathcal {A}$ with inverse s;

(ii) $\Vert s-e-x\Vert \le \frac{C_P^2\Vert x\Vert ^2}{(1-(C_P\Vert x\Vert )^q)^{1/q}}$;

(iii) $|\phi (x)|<1$ for all complex homomorphism $\phi$ on $\mathcal {A}$.

Proof

(i) It follows precisely as in [27, Theorem 10.7 (a)], with

$$\begin{aligned} \Vert x^m+x^{m+1}+\ldots +x^n\Vert ^q\le \sum _{j=m}^n\Vert x^j\Vert ^q\le \sum _{j=m}^n(C_P\Vert x\Vert )^{qj}, \end{aligned}$$

which goes to 0 since the series converges. This proves that the partial sums $s_n=e+x+x^2+\ldots +x^n$ form a Cauchy sequence in $\mathcal {A}$. Moreover, we also have $\Vert x^n\Vert \rightarrow 0$ as $n\rightarrow +\infty$ because

$$\begin{aligned} \Vert x^n\Vert ^q\le C_P^{nq}\Vert x\Vert ^{nq}\rightarrow 0 \end{aligned}$$

since $C_P\Vert x\Vert <1$ So, all the ingredients used to prove (i) are still valid.

The proof of (ii) goes exactly as that of [27, Theorem 10.7 (b)], with the difference that

$$\begin{aligned} \Vert s-e-x\Vert ^q=\Vert x^2+x^3+\ldots \Vert ^q\le \sum _{j=2}^\infty (C_P)^{jq}\Vert x\Vert ^{jq}=\frac{C_P^{2q}\Vert x\Vert ^{2q}}{1-(C_P\Vert x\Vert )^q}. \end{aligned}$$

(iii) It is proved verbatim as in [27, Theorem 10.7 (c)]. $\square$

We denote with $G(\mathcal {A})$ the group of invertible elements of $\mathcal {A}$. If $x\in \mathcal {A}$, the spectrum of x is defined exactly as in the Banach setting as

$$\begin{aligned} \sigma (x)=\{\lambda \in \mathbb {C}\ : \ \lambda e-x \ \text{ is } \text{ not } \text{ invertible }\}. \end{aligned}$$

$\mathbb {C}\setminus \sigma (x)$ is called the resolvent of x and $\rho (x)=\sup _{\lambda \in \sigma (x)}|\lambda |$ is the spectral radius of x. The following result generalizes [27, Theorem 10.11] to the quasi-Banach setting, and its proof is also a straightforward generalization.

Theorem A.6

Let $\mathcal {A}$ be a quasi- Banach algebra, $x\in G(\mathcal {A})$ and $h\in \mathcal {A}$ be such that $\Vert h\Vert <\frac{1}{2C_P^2}\Vert x^{-1}\Vert ^{-1}$. Then, $x+h\in G(\mathcal {A})$ and

$$\begin{aligned} \Vert (x+h)^{-1}-x^{-1}+x^{-1}hx^{-1}\Vert \le C_P^4\Vert x^{-1}\Vert ^3\Vert h\Vert ^2. \end{aligned}$$

Proof

Since $\Vert h\Vert <\frac{1}{2C_P^2}\Vert x^{-1}\Vert ^{-1}$,

$$\begin{aligned} \Vert x^{-1}h\Vert \le C_P\Vert x^{-1}\Vert \Vert h\Vert<C_P\frac{1}{2C_P^2}\Vert x^{-1}\Vert ^{-1}\Vert x^{-1}\Vert =\frac{1}{2C_P}<\frac{1}{C_P} \end{aligned}$$

By Theorem A.5, $x^{-1}h$ is invertible in $\mathcal {A}$ and

$$\begin{aligned} \begin{aligned} \Vert (x+h)^{-1}-x^{-1}x^{-1}hx^{-1}\Vert&=\Vert (e+x^{-1}h)^{-1}-e+x^{-1}h\Vert \Vert x^{-1}\Vert \\&\le \frac{C_P^2\Vert x^{-1}h\Vert ^2}{(1-C_P^q\Vert x^{-1}h\Vert ^q)^{1/q}}\Vert x^{-1}\Vert \\&\le C_P^2\Vert x^{-1}h\Vert ^2\Vert x^{-1}\Vert \le C_P^4\Vert x^{-1}\Vert ^3\Vert h\Vert ^2. \end{aligned} \end{aligned}$$

$\square$

As a consequence, $G(\mathcal {A})$ is open and $x\mapsto x^{-1}$ is a homomorphism of $G(\mathcal {A})$ onto itself, cf. [27, Theorem 1.12]. It is also immediate to verify that [27, Theorem 1.13] generalizes with the same statement, and the upper bound for $\rho (x)$ changes to

$$\begin{aligned} \rho (x)\le C_P\Vert x\Vert \qquad x\in \mathcal {A}, \end{aligned}$$

in particular the spectral radius formula holds:

$$\begin{aligned} \rho (x)=\lim _{n\rightarrow +\infty }\Vert x^n\Vert ^{1/n}=\inf _{n\ge 1}\Vert x^n\Vert ^{1/n} \end{aligned}$$

and the proof of the [27, Theorem 10.14] extends to the quasi-Banach setting.

Theorem A.7

(Gelfand-Mazur) If $\mathcal {A}$ is a quasi-Banach algebra and $G(\mathcal {A})=\mathcal {A}\setminus \{0\}$, then $\mathcal {A}$ is (isometrically) isomorphic to $\mathbb {C}$.

Remark A.8

The condition $\Vert e\Vert =1$ serves in the proof of Theorem A.7 to prove that the isomorphism $\lambda :\mathcal {A}\rightarrow \mathbb {C}$ of the Theorem of Gelfand-Mazur is an isometry. If $\Vert e\Vert >0$, then $\lambda$ is a quasi-isometry, as $|\lambda (x)|=\Vert e\Vert \Vert x\Vert$ for all $x\in \mathcal {A}$. Condition A.3 (v) is barely used in this part of Banach quasi-algebras and, exactly as condition (42) with $C_P>0$, it has a minor impact on the validity of the Banach setting results.

Definition A.9

Let $\mathcal {A}$ be a commutative complex quasi-Banach algebra. A linear subspace $\mathcal {J}\subseteq \mathcal {A}$ is an ideal of $\mathcal {A}$ if $x\cdot y\in \mathcal {J}$ for all $x\in \mathcal {A}$ and all $y\in \mathcal {J}$. $\mathcal {J}$ is proper if $\mathcal {J}\ne \mathcal {A}$ and it is maximal if it proper and it is not contained in any larger proper ideal.

[27, Proposition 11.2] and [27, Theorem 11.3] extend trivially to the quasi-Banach setting.

Let $\mathcal {J}$ be a closed and proper ideal of $\mathcal {A}$. Let $\pi :\mathcal {A}\rightarrow \mathcal {A}/\mathcal {J}$ be the quotient map $\pi (a)=a+\mathcal {J}$ ($a\in \mathcal {A}$). Define

$$\begin{aligned} \Vert a+\mathcal {J}\Vert :=\inf _{y\in J}\Vert a+y\Vert . \end{aligned}$$

Then, $\Vert \cdot \Vert$ defines a complex quasi-Banach algebra structure on $\mathcal {A}/\mathcal {J}$. In fact, the product on $\mathcal {A}/\mathcal {J}$ is defined precisely as in the Banach setting. Moreover, $\Vert \pi (x)\Vert \le \Vert x\Vert$ since $0\in \mathcal {J}$, so $\pi$ is continuous with respect to the quasi-norm topologies. A slightly modification of the proof for the Banach setting leads to the inequality

$$\begin{aligned} \Vert \pi (x)\pi (y)\Vert \le C_P\Vert \pi (x)\Vert \Vert \pi (y)\Vert \qquad \forall \pi (x),\pi (y)\in \mathcal {A}/\mathcal {J}. \end{aligned}$$

Finally, $\Vert \pi (e)\Vert =\Vert \pi (e)\pi (e)\Vert \le C_P\Vert \pi (e)\Vert ^2$, which implies that $\Vert \pi (e)\Vert \ge 1/C_P$. If $C_P=1$, this implies that $\Vert \pi (e)\Vert \ge 1$ and the other inequality follows trivially by the continuity of $\pi$. If $C_P>0$, then we have

$$\begin{aligned} \frac{1}{C_P}\le \Vert \pi (e)\Vert \le 1. \end{aligned}$$

For this reason, when dealing with quotients quasi-algebras, condition (v) of Definition A.3 can be replaced by $\frac{1}{C_P}\le \Vert {\pi }(e)\Vert \le 1$.

For our purposes $C_P=1$ and so also $\Vert \pi (e)\Vert =1$.

Theorem A.10

Let $\mathcal {A}$ be a commutative quasi-Banach algebra and

$$\begin{aligned} \widehat{\mathcal {A}}:=\{\phi :\mathcal {A}\rightarrow \mathbb {C}, \ complex \ homomorphism\}. \end{aligned}$$

Then, (i) every maximal ideal of $\mathcal {A}$ is the kernel of some $h\in \widehat{\mathcal {A}}$,

(ii) if $h\in \widehat{\mathcal {A}}$, $\ker (h)$ is a maximal ideal of $\mathcal {A}$,

(iii) $x\in \mathcal {A}$ is invertible if and only if $h(x)\ne 0$ for all $h\in \widehat{\mathcal {A}}$,

(iv) $x\in \mathcal {A}$ is invertible if and only if x lies in no proper ideal of $\mathcal {A}$,

(v) $\lambda \in \sigma (x)$ if and only if $h(x)=\lambda$ for some $h\in \widehat{\mathcal {A}}$.

Proof

Is just a readjustment of the proof of [27, Theorem 11.5]. $\square$

Definition A.11

Let $\mathcal {A}$ be a commutative quasi-Banach algebra and $\widehat{\mathcal {A}}$ be the set of the complex homomorphism of $\mathcal {A}$. The Gelfand transform of $x\in \mathcal {A}$ is the mapping ${\hat{x}}:\Delta \rightarrow \mathbb {C}$ defined for all $h\in \widehat{\mathcal {A}}$ as

$$\begin{aligned} {\hat{x}}(h)=h(x). \end{aligned}$$

Corollary A.12

Let $\mathcal {A}$ be a commutative quasi-Banach algebra. Then, $x\in \mathcal {A}$ is invertible if and only if ${\hat{x}}(h)\ne 0$ for all $h\in \widehat{\mathcal {A}}$.

Proof

It follows directly by Theorem A.10 (iii). $\square$

Following the pattern of Section 24 in [4], one can infer that the representation theory for quasi-Banach algebras goes exactly the same as for Banach algebras, since the main ingredients are the algebraic properties, the closedness criteria and the continuity of the representations. We then restate the same Lemmata 8.7, 8.8 and 8.9 in [24] in our setting as follows.

Let $\mathcal {A}$ be a quasi-Banach algebra with identity and $\mathcal {M}\subseteq \mathcal {A}$ a closed left ideal. Then $\mathcal {A}$ acts on the quasi-Banach space $\mathcal {A}/\mathcal {M}$ by the left regular representation

$$\begin{aligned} \pi _{\mathcal {M}}(a)\tilde{x}=\widetilde{ax},\quad a\in \mathcal {A}, \,\tilde{x}\in \mathcal {A}/\mathcal {M}, \end{aligned}$$

(43)

where $\tilde{x}$ is the equivalence class of x in $\mathcal {A}/\mathcal {M}$.

Lemma A.13

If $\mathcal {M}$ is a maximal left ideal of a quasi-Banach algebra $\mathcal {A}$, then $\pi _{\mathcal {M}}$ is algebraically irreducible. That is,

$$\begin{aligned} \{\pi _\mathcal {M}(a)\tilde{x}\,: \,a\in \mathcal {A}\}=\mathcal {A}/\mathcal {M}, \end{aligned}$$

for every $\tilde{x}\not =0$.

Lemma A.14

Let $\mathcal {A}$ be a quasi-Banach algebra with identity. An element $\mathcal {A}$ is left-invertible (right-invertible) if and only if $\pi _{\mathcal {M}}(a)$ is invertible for every maximal left (right) ideal $\mathcal {M}\subseteq \mathcal {A}$.

Lemma A.15

(Schur’s Lemma for quasi-Banach space representations) Assume that $\pi :\mathcal {A}\rightarrow \mathcal {B}(X)$ is an algebraically irreducible representation of $\mathcal {A}$ on a quasi-Banach space X. If $T\in \mathcal {B}(X)$ and $T\pi (a)=\pi (a)T$ for all $a\in \mathcal {A}$, then T is a multiple of the identity operator $\textrm{Id}$ on X.

1.2 A.2. The quasi-Banch algebras $\mathcal {B}$

Observe that, for $0<q\le 1$,

$$\begin{aligned} \ell ^q_{v_s}({\mathbb {Z}^{2d}})\hookrightarrow \ell ^2({\mathbb {Z}^{2d}}),\quad s\ge 0. \end{aligned}$$

(44)

(continuous embedding).

Let $\mathcal {D}:=\{a\in \ell ^2(\mathbb {Z}^d): \, \mathcal {F}a\in L^\infty (\mathbb {T}^d)\}$ be the Banach algebra with the norm $\Vert a\Vert _{\mathcal {D}}:=\Vert \mathcal {F}a\Vert _\infty ,$ where

$$\begin{aligned} \mathcal {F}a(\xi )=\sum _{n\in \mathbb {Z}^d} a(n)e^{2\pi i n\xi }. \end{aligned}$$

(45)

Lemma A.16

$\mathcal {B}$ is continuously embedded in $\mathcal {D}$ and $\Vert a\Vert _{\mathcal {D}}\le \Vert a\Vert _{\mathcal {B}}$.

Proof

Since $\ell ^q_{v_s}(\mathbb {Z}^d)\hookrightarrow \ell ^1(\mathbb {Z}^d)$ with $\Vert a\Vert _{\ell ^1}\le \Vert a\Vert _{\ell ^q_{v_s}}$ and $\ell ^1(\mathbb {Z}^d)\hookrightarrow \mathcal {D}$ with $\Vert a\Vert _{\mathcal {D}}\le \Vert a\Vert _{\ell ^1}$, the result immediately follows. $\square$

We recall a list of Lemmata from [24]. Namely,

Lemma A.17

(Lemma 8.3 [24]) If $b\in \mathcal {D}$ and $|a|\le b$ then $a\in \mathcal {D}$ and $\Vert a\Vert _{\mathcal {D}}\le \Vert b\Vert _{\mathcal {D}}$.

Lemma A.18

(Lemma 8.4 [24]) Let a be a sequence on $\mathbb {Z}^d$ such that $\mathcal {F}|a|$ is well defined. Then

$$\begin{aligned} \Vert a\Vert _1=\Vert \mathcal {F}|a|\Vert _\infty . \end{aligned}$$

(46)

Proposition A.19

(i) The Gelfand transform of $a\in \ell ^1(\mathbb {Z}^d)$ coincides with the Fourier series $\mathcal {F}a$ in (45).

(ii) The convolution operator $C_a b=a*b$ for $a\in \ell ^1(\mathbb {Z}^d)$ is invertible if and only if the Fourier series (45) does not vanish at any $\xi \in \mathbb {T}^d$.

(iii) If $a\in \mathcal {B}$, then the restriction of the Gelfand transform of a to $\mathbb {T}^d$ is the Fourier series $\mathcal {F}a$ of a.

Proof

For Items (i) and (ii) see [24]. Item (iii) follows from the inclusion $\ell ^q_{v_s}(\mathbb {Z}^d)\subseteq \ell ^1(\mathbb {Z}^d)$, $0<q\le 1$. We know that if $\mathcal {B}\subseteq \ell ^1$, then $\mathbb {T}^d\subseteq \widehat{\mathcal {B}}$ and the Fourier transform is the restriction of the Gelfand transform on $\mathbb {T}^d$, so (iii) hold if $\mathbb {T}^d\simeq \widehat{\mathcal {B}}$. $\square$

As a consequence of Corollary A.12 and Proposition A.19, we have

Theorem A.20

Assume that ${\widehat{B}}\simeq \mathbb {T}^d$. An element $a\in \mathcal {B}$ is invertible if and only if its Fourier series $\mathcal {F}a$ does not vanish at any point.