The Gabor wave front set in spaces of ultradifferentiable functions

TL;DR

This paper introduces the $ ext{Gabor}$ wave front set within ultradifferentiable function spaces, establishing their equivalence and exploring applications to differential and pseudo-differential operators.

Contribution

It defines the $ ext{Gabor}$ wave front set in ultradifferentiable spaces and proves its equivalence with the $ ext{omega}$-wave front set, extending microlocal analysis tools.

Findings

The $ ext{Gabor}$ and $ ext{omega}$-wave front sets coincide.

The framework applies to differential and pseudo-differential operators.

Extension of wave front set concepts to ultradifferentiable function spaces.

Abstract

Given a non-quasianalytic subadditive weight function $\omega$ we consider the weighted Schwartz space $\mathcal{S}_\omega$ and the short-time Fourier transform on $\mathcal{S}_\omega$, $\mathcal{S}'_\omega$ and on the related modulation spaces with exponential weights. In this setting we define the $\omega$-wave front set $WF'_\omega(u)$ and the Gabor $\omega$-wave front set $WF^G_\omega(u)$ of $u\in\mathcal{S}'_{\omega}$, and we prove that they coincide. Finally we look at applications of this wave front set for operators of differential and pseudo-differential type.

Equations

$$\underset{x\in {\mathbb{R}}^d}{\sup }{e}^{\lambda \omega (x)}|{\partial }^{\alpha }f(x)|<+\infty ,\underset{\xi \in {\mathbb{R}}^d}{\sup }{e}^{\lambda \omega (\xi )}|{\partial }^{\alpha }\hat{f}(\xi )|<+\infty ,$$

$$\mathcal{F}(f)=\hat{f}(\xi )={\int }_{{\mathbb{R}}^d}{e}^{-i⟨x,\xi ⟩}f(x)dx,$$

$${\phi }_{\omega }^{\ast }(s):=\underset{t\ge 0}{\sup }\{ts-{\phi }_{\omega }(t)\}$$

$$p_{\lambda ,\mu }(u):=\underset{\alpha ,\beta \in {\mathbb{N}}_0^d}{\sup }\underset{x\in {\mathbb{R}}^d}{\sup }|x^{\beta }{D}^{\alpha }u(x)|e^{-\lambda {\phi }_{\omega }^{\ast }\left(\frac{|\alpha |}{\lambda }\right)-\mu {\phi }_{\omega }^{\ast }\left(\frac{|\beta |}{\mu }\right)},$$

$$q_{\lambda ,\mu }(u):=\underset{\alpha \in {\mathbb{N}}_0^d}{\sup }\underset{x\in {\mathbb{R}}^d}{\sup }|D^{\alpha }u(x)|e^{-\lambda {\phi }_{\omega }^{\ast }\left(\frac{|\alpha |}{\lambda }\right)+\mu \omega (x)},\lambda ,\mu >0,$$

$$T_{x}f(y)=f(y-x),M_{\xi }f(y)=e^{i⟨y,\xi ⟩}f(y),\Pi (z)f(y)=M_{\xi }{T}_{x}f(y)=e^{i⟨y,\xi ⟩}f(y-x),$$

$$V_{\phi }f(z):=$$

$$=$$

$$⟨\psi ,\phi ⟩f(y)=\frac{1}{(2\pi {)}^d}{\int }_{{\mathbb{R}}^{2d}}{V}_{\phi }f(z)(\Pi (z)\psi )(y)dz.$$

$$f(y)=\frac{1}{(2\pi {)}^d\Vert \phi {\Vert }_{L^2}^2}{\int }_{{\mathbb{R}}^{2d}}{V}_{\phi }f(z)(\Pi (z)\phi )(y)dz.$$

$$|V_{\phi }f(z)|\le c{e}^{\lambda \omega (z)}\forall z\in {\mathbb{R}}^{2d}.$$

$$|F(z)|\le C_{\lambda }{e}^{-\lambda \omega (z)}\forall z\in {\mathbb{R}}^{2d}.$$

$$f(y):={\int }_{{\mathbb{R}}^{2d}}F(z)(\Pi (z)\phi )(y)dz$$

$$|V_{\phi }f(z)|\le C_{\lambda }{e}^{-\lambda \omega (z)}\forall z\in {\mathbb{R}}^{2d};$$

$$\widehat{V_{\phi }f}(\eta ,y)=(2\pi {)}^d{e}^{i⟨\eta ,y⟩}f(-y)\overline{\hat{\phi }(\eta )}\forall (\eta ,y)\in {\mathbb{R}}^{2d}.$$

$$V_{\phi }:{\mathcal{S}}_{\omega }({\mathbb{R}}^d)⟶{\mathcal{S}}_{\omega }({\mathbb{R}}^{2d})$$

$$f_n⟶f\in {\mathcal{S}}_{\omega }({\mathbb{R}}^d)\text{mbox}in{\mathcal{S}}_{\omega }({\mathbb{R}}^d)$$

$$e^{i⟨\eta ,y⟩}{f}_n(-y)\overline{\hat{\phi }(\eta )}⟶e^{i⟨\eta ,y⟩}f(-y)\overline{\hat{\phi }(\eta )}\text{mbox}in{\mathcal{S}}_{\omega }({\mathbb{R}}^{2d})$$

$$\widehat{V_{\phi }{f}_n}\to \widehat{V_{\phi }f}\text{mbox}in{\mathcal{S}}_{\omega }({\mathbb{R}}^{2d}).$$

$$V_{\phi }{f}_n\to V_{\phi }f\text{mbox}in{\mathcal{S}}_{\omega }({\mathbb{R}}^{2d}).$$

$$\Vert V_{\phi }f{\Vert }_{\omega ,\lambda }:=\underset{z\in {\mathbb{R}}^{2d}}{\sup }|V_{\phi }f(z)|e^{\lambda \omega (z)},$$

$${\tilde{\mathcal{S}}}_{\omega }({\mathbb{R}}^d):=\{f\in \mathcal{S}({\mathbb{R}}^d):\Vert V_{\phi }f{\Vert }_{\omega ,\lambda }<+\infty \forall \lambda >0\}.$$

$$e^{-\lambda {\phi }_{\omega }^{\ast }\left(\frac{|\alpha |}{\lambda }\right)}{e}^{-\mu {\phi }_{\omega }^{\ast }\left(\frac{|\beta |}{\mu }\right)}|y^{\beta }{D}_y^{\alpha }f(y)|$$

$$|y{|}^{|\beta |}{e}^{-\mu {\phi }_{\omega }^{\ast }\left(\frac{|\beta |}{\mu }\right)}\le e^{\mu \omega (y)},\text{mbox}forally\in {\mathbb{R}}^d.$$

$$2^{|\alpha |}{e}^{-\lambda {\phi }_{\omega }^{\ast }\left(\frac{|\alpha |}{\lambda }\right)}\le e^{3\lambda }{e}^{-3\lambda {\phi }_{\omega }^{\ast }\left(\frac{|\alpha |}{3\lambda }\right)}.$$

$$e^{-\lambda {\phi }_{\omega }^{\ast }\left(\frac{|\alpha |}{\lambda }\right)-\mu {\phi }_{\omega }^{\ast }\left(\frac{|\beta |}{\mu }\right)}|y^{\beta }{D}_y^{\alpha }f(y)|\le$$

$$|D_y^{\gamma }\phi (y)|e^{\mu \omega (y)}\le C_{\lambda ,\mu }{e}^{\lambda {\phi }_{\omega }^{\ast }\left(\frac{|\gamma |}{\lambda }\right)}.$$

$$e^{-\lambda {\phi }_{\omega }^{\ast }\left(\frac{|\alpha |}{\lambda }\right)-\mu {\phi }_{\omega }^{\ast }\left(\frac{|\beta |}{\mu }\right)}|y^{\beta }{D}_y^{\alpha }f(y)|\le$$

$$e^{-\lambda {\phi }_{\omega }^{\ast }\left(\frac{|\alpha |}{\lambda }\right)-\mu {\phi }_{\omega }^{\ast }\left(\frac{|\beta |}{\mu }\right)}|y^{\beta }{D}_y^{\alpha }f(y)|$$

$$\le C_{\mu ,\lambda }^{\prime }\Vert V_{\phi }f{\Vert }_{\omega ,\mu +3\lambda +1},$$

$$|F(z)|\le c{e}^{\lambda \omega (z)}\forall z\in {\mathbb{R}}^{2d},$$

Full text

1. Introduction

The wave front set is a basic concept in the theory of linear partial differential operators. It deals with the analysis of singularities of a function (or distribution), and in the classical Schwartz distributions theory it was originally defined in [H-1]. The idea is that, if a distribution $u\in\mathcal{D}^{\prime}(\mathbb{R}^{d})$ coincides with a $C^{\infty}$ function in a neighborhood of a certain point $x_{0}\in\mathbb{R}^{d}$, then there exists a cut-off function $\varphi\in\mathcal{D}(\mathbb{R}^{d})$ (i.e., $\varphi\in C^{\infty}(\mathbb{R}^{d})$ with compact support) such that $\varphi u\in\mathcal{D}(\mathbb{R}^{d})$, and consequently $\widehat{\varphi u}$ is a rapidly decreasing function. If $u$ is not $C^{\infty}$ at $x_{0}$, then $\widehat{\varphi u}$ does not decrease rapidly at least in some directions, and these directions are responsible for the absence of regularity of $u$ at $x_{0}$. The wave front set collects all the points $(x_{0},\xi_{0})$, with $\xi_{0}\neq 0$, where, roughly speaking, the distribution $u$ is not $C^{\infty}$ at $x_{0}$ due (on the Fourier transform side) to the absence of rapid decreasing in the direction $\xi_{0}$. The wave front set is then a subset of $\mathbb{R}^{d}\times(\mathbb{R}^{d}\setminus\{0\})$, is computed for a distribution $u$ and has to do with the analysis of the points where $u$ is not smooth in connection with the directions where absence of smoothness is shown by the Fourier transform of $u$. In the case described above, the distribution space is $\mathcal{D}^{\prime}$, and ‘smooth’ means $C^{\infty}$.

Some very natural questions arise. First, the spaces $\mathcal{D}^{\prime}$ and $C^{\infty}$ could be replaced by other spaces of distributions connected to other concepts of ‘smoothness’; in this frame we refer for example to [R], [BJJ], [AJO-1], [AJO-2], where wave front sets connected with Gevrey and ultradifferentiable type regularity are considered. Moreover, in the case of $\mathcal{D}^{\prime}$ and $C^{\infty}$ the smoothness is intended in a local sense, but also some kind of global regularity could be considered, see for example [H-2], [CS], [RW], [N], [SW-2].

The wave front set, moreover, has very important applications in the study of propagation of singularities for partial differential (or more generally for pseudodifferential) operators. In this frame, different classes of pseudodifferential operators lead to corresponding variants of the wave front set, adapted to the class under consideration. Among the vast literature in this field we refer for example to [H-3] for the $C^{\infty}$ case, to [R] for the Gevrey case, and to [SW-1], [CW] for the case of global wave front set defined in the spirit of the present paper.

Time-frequency analysis is a field of research that in the last decades has had a very big growth, with the development of many new techniques. One of the basic ideas of time-frequency analysis is the simultaneous analysis of a function (or distribution) with respect to variables and covariables, in order to quantify the energy of a signal at some time $x_{0}$ and some frequency $\xi_{0}$. Since the wave front set has to do with a simultaneous analysis of points (variables) and directions (covariables), it is very natural to try to apply methods of time-frequency analysis in connection with the wave front set. The work [RW] is a very interesting contribution in this direction.

The present paper deals with a global wave front set, in the spirit of [H-2], treated with techniques from time-frequency analysis, following ideas from [RW]. In particular, we study the case of ultradifferentiable functions in the sense of [B], [BMT], focusing on the space $\mathcal{S}_{\omega}(\mathbb{R}^{d})$, defined as the space of functions $f\in\mathcal{S}(\mathbb{R}^{d})$ such that for every $\lambda>0$ and $\alpha\in\mathbb{N}^{d}_{0}$ we have

[TABLE]

where $\omega$ is a non-quasianalytic subadditive weight, cf. Definition 2.1. The spaces $\mathcal{S}_{\omega}(\mathbb{R}^{d})$, together with the corresponding (ultra)distribution space $\mathcal{S}^{\prime}_{\omega}(\mathbb{R}^{d})$, play a role similar to $\mathcal{S}(\mathbb{R}^{d})$ and $\mathcal{S}^{\prime}(\mathbb{R}^{d})$ in the classical Schwartz frame. We study in this paper a global wave front set adapted to $\mathcal{S}_{\omega}$ regularity, giving two different definitions; one is based on Gabor transform and the other is related to Gabor frames. We show that these definitions are equivalent. Moreover, we give applications of this global wave front set to pseudodifferential operators, to partial differential operators with polynomial coefficients, to localization operators, and we analyze some examples. The techniques are related with time-frequency analysis, in particular to Gabor transform, Gabor frames and modulation spaces.

The paper is organized as follows. In Section 2 we present basic definitions and consider time-frequency analysis on ultradifferentiable spaces; we revise some known properties and prove other results that are needed in the paper, but that we could not find in the literature. In Section 3 we give the two definitions of wave front set and prove that they are equivalent. Moreover we show that the global wave front set of a distribution $u\in\mathcal{S}^{\prime}_{\omega}(\mathbb{R}^{d})$ is empty if and only if $u\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$, and that the global wave front set is not affected by translations and modulations, that are the basic operators of time-frequency analysis. In Section 4 we present some applications about the action of operators of differential and pseudodifferential type on the wave front set, and finally in Section 5 we analyze some examples.

2. Preliminaries and the short-time Fourier transform in $\mathcal{S}_{\omega}(\mathbb{R}^{d})$

Given a function $f\in L^{1}(\mathbb{R}^{d})$, the Fourier transform of $f$ is defined as

[TABLE]

with standard extensions to more general spaces of functions and distributions.

Definition 2.1**.**

A non-quasianalytic subadditive weight function is a continuous increasing function $\omega:\ [0,+\infty)\to[0,+\infty)$ satisfying the following properties:

• $(\alpha)$

$\omega(t_{1}+t_{2})\leq\omega(t_{1})+\omega(t_{2})\quad\forall t_{1},t_{2}\geq 0$;

• $(\beta)$

$\displaystyle\int_{1}^{+\infty}\frac{\omega(t)}{t^{2}}dt<+\infty$;

• $(\gamma)$

$\log t=o(\omega(t))$* when $t\to+\infty$;*

• $(\delta)$

$\varphi_{\omega}(t):=\omega(e^{t})$* is convex.*

We then define $\omega(\zeta):=\omega(|\zeta|)$ for $\zeta\in\mathbb{C}^{d}$.

We denote by $\varphi^{*}_{\omega}$ the Young conjugate of $\varphi_{\omega}$, defined by

[TABLE]

and recall that $\varphi^{*}_{\omega}$ is convex and increasing, $\varphi^{**}_{\omega}=\varphi_{\omega}$ and $\varphi^{*}_{\omega}(s)/s$ is increasing (up to assume, without any loss of generality, that $\omega|_{[0,1]}\equiv 0$).

Definition 2.2**.**

We define $\mathcal{S}_{\omega}(\mathbb{R}^{d})$ as the set of all $u\in\mathcal{S}(\mathbb{R}^{d})$ such that

• (i)

$\displaystyle\forall\lambda>0,\alpha\in\mathbb{N}_{0}^{d}:\ \sup_{\mathbb{R}^{d}}e^{\lambda\omega(x)}|D^{\alpha}u(x)|<+\infty$,

• (ii)

$\displaystyle\forall\lambda>0,\alpha\in\mathbb{N}_{0}^{d}:\ \sup_{\mathbb{R}^{d}}e^{\lambda\omega(\xi)}|D^{\alpha}\widehat{u}(\xi)|<+\infty$,

where $\mathbb{N}_{0}:=\mathbb{N}\cup\{0\}$ and $D^{\alpha}=(-i)^{|\alpha|}\partial^{\alpha}$.

As usual, the corresponding dual space is denoted by $\mathcal{S}^{\prime}_{\omega}(\mathbb{R}^{d})$ and is the set of all linear and continuous functionals $u:\,\mathcal{S}_{\omega}(\mathbb{R}^{d})\to\mathbb{C}$. An element of $\mathcal{S}^{\prime}_{\omega}(\mathbb{R}^{d})$ is called an $\omega$-tempered distribution.

In [BJO, Thm. 4.8] we provided the space $\mathcal{S}_{\omega}(\mathbb{R}^{d})$ with different equivalent systems of seminorms. For example, for $u\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$, the family of seminorms

[TABLE]

for $\lambda,\mu>0$. On the other hand, it is not difficult to see (using, for instance, [BJO, Lemma 4.7(ii)]) that the family of seminorms

[TABLE]

defines another equivalent system of seminorms for $\mathcal{S}_{\omega}(\mathbb{R}^{d})$.

We recall that $\mathcal{S}_{\omega}(\mathbb{R}^{d})\subseteq\mathcal{S}(\mathbb{R}^{d})$ and for their correspondent dual spaces we have the inclusion $\mathcal{S}^{\prime}(\mathbb{R}^{d})\subseteq\mathcal{S}^{\prime}_{\omega}(\mathbb{R}^{d})$.

Let us denote by $T_{x}$, $M_{\xi}$ and $\Pi(z)$, respectively, the translation, the modulation and the phase-space shift operators, defined by

[TABLE]

for $x,y,\xi\in\mathbb{R}^{d}$ and $z=(x,\xi)$.

Definition 2.3**.**

For a window function $\varphi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})\setminus\{0\}$, the short-time Fourier transform (briefly STFT) of $f\in\mathcal{S}^{\prime}_{\omega}(\mathbb{R}^{d})$ is defined, for $z=(x,\xi)\in\mathbb{R}^{2d}$, by:

[TABLE]

where the bracket $\langle\cdot,\cdot\rangle$ in (2.3) and the integral in (2.4) denote the conjugate linear action of $\mathcal{S}^{\prime}_{\omega}$ on $\mathcal{S}_{\omega}$, consistent with the inner product $\langle\cdot,\cdot\rangle_{L^{2}}$.

By [GZ, Lemma 1.1], for $f,\varphi,\psi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$ we have the following inversion formula:

[TABLE]

In particular, for $\psi=\varphi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})\setminus\{0\}$:

[TABLE]

We recall, from [GZ], the following results:

Theorem 2.4**.**

Let $\varphi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})\setminus\{0\}$ and $f\in\mathcal{S}^{\prime}_{\omega}(\mathbb{R}^{d})$. Then $V_{\varphi}f$ is continuous and there are constants $c,\lambda>0$ such that

[TABLE]

Proposition 2.5**.**

Let $\varphi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})\setminus\{0\}$ and assume that $F:\ \mathbb{R}^{2d}\to\mathbb{C}$ is a measurable function that satisfies that for all $\lambda>0$ there is a constant $C_{\lambda}>0$ such that

[TABLE]

Then

[TABLE]

defines a function $f\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$.

Theorem 2.6**.**

Let $\varphi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})\setminus\{0\}$. Then, for $f\in\mathcal{S}^{\prime}_{\omega}(\mathbb{R}^{d})$, the following are equivalent:

• (i)

$f\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$;

• (ii)

for all $\lambda>0$ there exists $C_{\lambda}>0$ such that

[TABLE]

• (iii)

$V_{\varphi}f\in\mathcal{S}_{\omega}(\mathbb{R}^{2d})$.

The following lemma is well known for functions in $\mathcal{S}(\mathbb{R}^{d})$, and hence in $\mathcal{S}_{\omega}(\mathbb{R}^{d}).$ So we omit its proof.

Lemma 2.7**.**

For $f,\varphi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$ we have that

[TABLE]

As a consequence, we can deduce the following result.

Proposition 2.8**.**

Let $\varphi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})\setminus\{0\}$. Then

[TABLE]

is continuous.

Proof.

Let us first remark that if $f\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$ then $V_{\varphi}f\in\mathcal{S}_{\omega}(\mathbb{R}^{2d})$ by Theorem 2.6.

Since $\mathcal{S}_{\omega}$ is a Fréchet space, to prove the continuity of $V_{\varphi}$ we consider a sequence $\{f_{n}\}_{n\in\mathbb{N}}\subset\mathcal{S}_{\omega}(\mathbb{R}^{d})$ such that

[TABLE]

and prove that $V_{\varphi}f_{n}\to V_{\varphi}f$ in $\mathcal{S}_{\omega}(\mathbb{R}^{2d})$.

Indeed, (2.8) implies that

[TABLE]

and hence, by Lemma 2.7,

[TABLE]

Applying the inverse Fourier transform, which is continuous on $\mathcal{S}_{\omega}$, we have that

[TABLE]

and the proof is complete. ∎

The short-time Fourier transform also provides a new equivalent system of seminorms for $\mathcal{S}_{\omega}(\mathbb{R}^{d})$

Proposition 2.9**.**

If $\varphi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})\setminus\{0\}$, then the collection of seminorms

[TABLE]

for $\lambda>0$, forms an equivalent system of seminorms for $\mathcal{S}_{\omega}(\mathbb{R}^{d})$.

Proof.

Set

[TABLE]

By Theorem 2.6 the sets $\tilde{\mathcal{S}}_{\omega}(\mathbb{R}^{d})$ and $\mathcal{S}_{\omega}(\mathbb{R}^{d})$ are equal. We have to prove that they have the same topology.

By the inversion formula (2.6) we have that, for $z=(x,\xi)\in\mathbb{R}^{2d}$ and $\lambda,\mu>0$,

[TABLE]

for some $C>0$.

The following property is known and can be found, for instance, in Lemma 4.7(i) of [BJO] (see also [FGJ]):

[TABLE]

Moreover, since the weight function $\omega$ is subadittive and $\varphi^{*}$ is convex, we can use, for instance, Proposition 2.1(e) of [BJ] to obtain

[TABLE]

Substituting (2.10) and (2.11) into (2), by the subadditivity of $\omega$ we have

[TABLE]

for some $C_{\lambda}>0$.

Since $\varphi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$, by (2.2), for every $\lambda,\mu>0$ there is a constant $C_{\lambda,\mu}>0$ such that for all $\gamma\in\mathbb{N}_{0}^{d}$ and $y\in\mathbb{R}^{d}$,

[TABLE]

From (2.13) with $3\lambda$ instead of $\lambda$ and $y-x$ instead of $y$, we have that for every $\mu,\lambda>0$ there exists a constant $C_{\mu,\lambda}>0$ such that

[TABLE]

By (2.10) we have $|\xi|^{|\alpha-\gamma|}\leq e^{3\lambda\omega(\xi)+3\lambda\varphi^{*}(\frac{|\alpha-\gamma|}{3\lambda})}$. So, from the convexity of $\varphi^{*}$, we obtain

[TABLE]

for $C^{\prime}_{\mu,\lambda}:=C_{\mu,\lambda}\int_{\mathbb{R}^{2d}}e^{-\omega(z)}dz$, which is finite by condition $(\gamma)$ of Definition 2.1.

It is easy to see that $\tilde{\mathcal{S}}_{\omega}(\mathbb{R}^{d})$ is a Fréchet space. Indeed, the estimate (2.15) implies that the identity operator $I:\ \tilde{\mathcal{S}}_{\omega}(\mathbb{R}^{d})\to\mathcal{S}_{\omega}(\mathbb{R}^{d})$ is continuous. Hence, any Cauchy sequence $\{f_{n}\}_{n\in\mathbb{N}}$ in $\tilde{\mathcal{S}}_{\omega}(\mathbb{R}^{d})$ is a Cauchy sequence in $\mathcal{S}_{\omega}(\mathbb{R}^{d})$. So, it converges in $\mathcal{S}_{\omega}(\mathbb{R}^{d})$ to some $f$ (because $\mathcal{S}_{\omega}(\mathbb{R}^{d})$ is complete). From Proposition 2.8, $\{V_{\varphi}f_{n}\}_{n\in\mathbb{N}}$ converges to $V_{\varphi}f$ in $\mathcal{S}_{\omega}(\mathbb{R}^{2d})$. Therefore, $\{f_{n}\}_{n\in\mathbb{N}}$ converges to $f$ in $\tilde{\mathcal{S}}_{\omega}(\mathbb{R}^{d}).$

We can apply the open mapping theorem to conclude that $I$ is an isomorphism and hence the two topologies on $\mathcal{S}_{\omega}(\mathbb{R}^{d})$ coincide. ∎

Now, we can prove the following

Proposition 2.10**.**

Assume that $\psi,\gamma\in\mathcal{S}_{\omega}(\mathbb{R}^{d})\setminus\{0\}$ with $\langle\psi,\gamma\rangle\neq 0$. Then the following assertions hold:

• (a)

If $F:\ \mathbb{R}^{2d}\to\mathbb{C}$ is a measurable function that satisfies, for some $c,\lambda>0$,

[TABLE]

then

[TABLE]

define an $\omega$-tempered distribution $f\in\mathcal{S}^{\prime}_{\omega}(\mathbb{R}^{d})$.

• (b)

In particular, if $F=V_{\psi}f$ for some $f\in\mathcal{S}^{\prime}_{\omega}(\mathbb{R}^{d})$, then the following inversion formula holds:

[TABLE]

Proof.

From (2.16) we have, for all $\varphi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$,

[TABLE]

for some $c^{\prime}>0$.

From Proposition 2.9 the inequality (2.18) implies that $f$ defines a continuous linear functional on $\mathcal{S}_{\omega}(\mathbb{R}^{d})$, i.e. $f\in\mathcal{S}^{\prime}_{\omega}(\mathbb{R}^{d})$. This proves $(a)$.

In particular, if $F=V_{\psi}f$ for some $f\in\mathcal{S}^{\prime}_{\omega}(\mathbb{R}^{d})$ then $F$ satisfies (2.16) for Theorem 2.4 and hence (2.17) defines an $\omega$-tempered distribution $\tilde{f}\in\mathcal{S}^{\prime}_{\omega}(\mathbb{R}^{d})$ given by

[TABLE]

However, from (2.5) we have that

[TABLE]

and then (see also [G, pg 43] for vector valued integrals)

[TABLE]

Therefore $f=\tilde{f}$ and $(b)$ is proved. ∎

Let us now recall the definition of the adjoint operator of $V_{\varphi}$. We consider, for $\varphi\in L^{2}(\mathbb{R}^{d})$, the operator

[TABLE]

defined by

[TABLE]

This is the adjoint operator of $V_{\varphi}:L^{2}(\mathbb{R}^{d})\to L^{2}(\mathbb{R}^{d})$ since, for all $F\in L^{2}(\mathbb{R}^{2d})$ and $h\in L^{2}(\mathbb{R}^{d})$,

[TABLE]

In particular, for $\varphi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$ and $F\in\mathcal{S}_{\omega}(\mathbb{R}^{2d})$ we can define the adjoint operator $V_{\varphi}^{*}F=A_{\varphi}F$. We observe that $V_{\varphi}^{*}F\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$. In fact, if $G(x,\xi,t):=F(x,\xi)\varphi(t-x)\in\mathcal{S}_{\omega}(\mathbb{R}^{3d})$, we can write $A_{\varphi}F$ as a partial Fourier transform:

[TABLE]

Then

[TABLE]

continuously.

Moreover, the inversion formula (2.5) gives, for $\varphi,\psi,f\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$ with $\langle\varphi,\psi\rangle\neq 0$,

[TABLE]

i.e.

[TABLE]

More in general, if $\varphi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})\setminus\{0\}$ and $F$ is a measurable function on $\mathbb{R}^{2d}$, we define the adjoint operator

[TABLE]

where the integral is interpreted, if necessary, in a weak sense, i.e.

[TABLE]

for $g\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$.

In particular, if $\varphi,\psi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})\setminus\{0\}$ with $\langle\varphi,\psi\rangle\neq 0$, by Theorem 2.4 and Proposition 2.10 we can define the adjoint operator (2.22) for $F=V_{\psi}f$ with $f\in\mathcal{S}^{\prime}_{\omega}(\mathbb{R}^{d})$ and obtain that, for all $g\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$,

[TABLE]

i.e.

[TABLE]

We can now prove the following proposition in a standard way.

Proposition 2.11**.**

Let $\varphi,\psi,\gamma\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$ with $\langle\gamma,\psi\rangle\neq 0$ and let $f\in\mathcal{S}^{\prime}_{\omega}(\mathbb{R}^{d})$. Then

[TABLE]

3. The $\omega$-Gabor wave front set

In this section, we consider a global wave front set defined in terms of rapid decay of the STFT in conical sets. After that, for a Gabor frame we define the Gabor wave front set, where conical sets are intersected with a lattice. We prove that these wave front sets coincide.

Definition 3.1**.**

Let $u\in\mathcal{S}^{\prime}_{\omega}(\mathbb{R}^{d})$ and $\varphi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})\setminus\{0\}$. We say that $z_{0}=(x_{0},\xi_{0})\in\mathbb{R}^{2d}\setminus\{0\}$ is not in the $\omega$-wave front set $\mathop{\rm WF}\nolimits^{\prime}_{\omega}(u)$ of $u$ if there exists an open conic set $\Gamma\subseteq\mathbb{R}^{2d}\setminus\{0\}$ containing $z_{0}$ and such that

[TABLE]

We observe that $\mathop{\rm WF}\nolimits^{\prime}_{\omega}(u)$ is a closed conic subset of $\mathbb{R}^{2d}\setminus\{0\}$. Moreover, it does not depend on the choice of the window function $\varphi$, as the following proposition shows.

Proposition 3.2**.**

Let $u\in\mathcal{S}^{\prime}_{\omega}(\mathbb{R}^{d})$, $\varphi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})\setminus\{0\}$ and $z_{0}\in\mathbb{R}^{2d}\setminus\{0\}$. Assume that there exists an open conic set $\Gamma\subseteq\mathbb{R}^{2d}\setminus\{0\}$ containing $z_{0}$ such that (3.25) is satisfied. Then, for any $\psi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})\setminus\{0\}$ and for any open conic set $\Gamma^{\prime}\subseteq\mathbb{R}^{2d}\setminus\{0\}$ containing $z_{0}$ and such that $\overline{\Gamma^{\prime}\cap S_{2d-1}}\subseteq\Gamma$, where $S_{2d-1}$ is the unit sphere in $\mathbb{R}^{2d}$, we have

[TABLE]

Proof.

From Proposition 2.11 we have that

[TABLE]

Moreover, since $\varphi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$, from Theorem 2.6 we have that for every $\mu>0$ there exists $C_{\mu}>0$ such that

[TABLE]

Then

[TABLE]

Let us choose $\varepsilon>0$ sufficiently small so that

[TABLE]

and hence, from (3.25), the subadditivity of $\omega$ and (3.28):

[TABLE]

On the other hand, from Theorem 2.4 and (3.28):

[TABLE]

for some $c^{\prime}>0$, if $\mu>\lambda+1$, since

[TABLE]

for some constant $A_{\varepsilon}>0$ which depends on the already fixed $\varepsilon>0$.

The arbitrariness of $\mu>\lambda+1$ in (3.31) implies that for every $\lambda^{\prime}>0$ there exists a constant $C_{\lambda^{\prime}}>0$ such that

[TABLE]

This gives the conclusion. ∎

Given $\alpha,\beta>0$, consider the lattice $\Lambda=\alpha\mathbb{Z}^{d}\times\beta\mathbb{Z}^{d}\subset\mathbb{R}^{2d}.$ For a window $\varphi\in L^{2}(\mathbb{R}^{d})\setminus\{0\}$ the collection $\{\Pi(\sigma)\varphi\}_{\sigma\in\Lambda}$ is called a Gabor frame for $L^{2}(\mathbb{R}^{d})$ provided there exists constants $A,B>0$ such that

[TABLE]

(see [G] for the analysis of the conditions on $\alpha$ and $\beta$ for which $\{\Pi(\sigma)\varphi\}_{\sigma\in\Lambda}$ is a Gabor frame). Now, we define the Gabor $\omega$-wave front set.

Definition 3.3**.**

Let $\varphi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})\setminus\{0\}$ and $\Lambda=\alpha_{0}\mathbb{Z}^{d}\times\beta_{0}\mathbb{Z}^{d}\subseteq\mathbb{R}^{2d}$ a lattice with $\alpha_{0},\beta_{0}>0$ sufficiently small so that $\{\Pi(\sigma)\varphi\}_{\sigma\in\Lambda}$ is a Gabor frame for $L^{2}(\mathbb{R}^{d})$. If $u\in\mathcal{S}^{\prime}_{\omega}(\mathbb{R}^{d})$, we say that $z_{0}\in\mathbb{R}^{2d}\setminus\{0\}$ is not in the Gabor $\omega$-wave front set $\mathop{\rm WF}\nolimits^{G}_{\omega}(u)$ of $u$ if there exists an open conic set $\Gamma\subset\mathbb{R}^{2d}\setminus\{0\}$ containing $z_{0}$ such that

[TABLE]

Our aim is to prove that $\mathop{\rm WF}\nolimits^{\prime}_{\omega}(u)=\mathop{\rm WF}\nolimits^{G}_{\omega}(u)$. We follow the lines of [RW] and we need some properties of modulation spaces that are already true in $\mathcal{S}(\mathbb{R}^{d})$ (see [G]), adapted to our setting.

We consider, for $\lambda\in\mathbb{R}\setminus\{0\}$,

[TABLE]

The weights $m_{\lambda}(z)$ are $v_{\lambda}$-moderate, in the sense that

[TABLE]

for every $\lambda\neq 0$ and $z_{1},z_{2}\in\mathbb{R}^{n}$. This is immediate from the subadditivity of $\omega$.

We denote, following [G], the weighted $L^{p,q}$ spaces by

[TABLE]

for $1\leq p,q<+\infty$, and

[TABLE]

for $1\leq p,q\leq+\infty$ with $p=+\infty$ or $q=+\infty$ respectively.

By [G, Lemma 11.1.2] these are Banach spaces for all $1\leq p,q\leq+\infty$. Moreover, for $F\in L^{p,q}_{m_{\lambda}}(\mathbb{R}^{2d})$ and $H\in L^{p^{\prime},q^{\prime}}_{1/{m_{\lambda}}}(\mathbb{R}^{2d})$, where $p^{\prime}$ and $q^{\prime}$ are the conjugate exponents of $p$ and $q$ respectively (i.e. $\frac{1}{p}+\frac{1}{p^{\prime}}=1$ if $1<p<+\infty$, $p^{\prime}=+\infty$ if $p=1$, $p^{\prime}=1$ if $p=+\infty$, and the same for $q$), then $F\cdot H\in L^{1}(\mathbb{R}^{2d})$ and

[TABLE]

If $1\leq p,q<+\infty$, the dual of $L^{p,q}_{m_{\lambda}}(\mathbb{R}^{2d})$ is given by $L^{p^{\prime},q^{\prime}}_{1/m_{\lambda}}(\mathbb{R}^{2d})$.

From [G, Proposition 11.1.3] we have the following Young inequality for weighted $L^{p,q}$ spaces. For $F\in L^{p,q}_{m_{\lambda}}$ and $G\in L^{1}_{v_{\lambda}}$,

[TABLE]

for some $C>0$.

Remark 3.4**.**

It is easy to see that for every $\lambda\in\mathbb{R}\setminus\{0\}$ and $1\leq p,q\leq+\infty$ we have

[TABLE]

Definition 3.5**.**

Let $\varphi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})\setminus\{0\}$, and $m_{\lambda}(z)$ as in (3.34) for some $\lambda\neq 0$. For $1\leq p,q\leq+\infty$, the modulation space $\boldsymbol{M}^{p,q}_{m_{\lambda}}(\mathbb{R}^{d})$ is defined by

[TABLE]

with norm $\|f\|_{\boldsymbol{M}^{p,q}_{m_{\lambda}}}=\|V_{\varphi}f\|_{L^{p,q}_{m_{\lambda}}}$. We denote then $\boldsymbol{M}^{p}_{m_{\lambda}}(\mathbb{R}^{d}):=\boldsymbol{M}^{p,p}_{m_{\lambda}}(\mathbb{R}^{d})$.

Observe that Definition 3.5 is similar to the definition of modulation spaces in [G]; the difference is that here $\boldsymbol{M}^{p,q}_{m_{\lambda}}(\mathbb{R}^{d})$ is a subset of $\mathcal{S}^{\prime}_{\omega}(\mathbb{R}^{d})$, and we take a window $\varphi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$, while in [G] the modulation space $M^{p,q}_{m}(\mathbb{R}^{d})$ is a subset of $\mathcal{S}^{\prime}(\mathbb{R}^{d})$ and the window belongs to $\mathcal{S}(\mathbb{R}^{d})$ (or a subset of $(M^{1}_{v})^{*}$ for a suitable weight $v$, in a suitable space of ‘special’ windows $\mathcal{S}_{\mathcal{C}}(\mathbb{R}^{d})$). Moreover, here we always need weights of exponential type. We refer to [T-1, T-2] for modulation spaces in the setting of Gelfand-Shilov spaces, among other type of spaces of ultradifferentiable functions and ultradistributions.

The definition of $\boldsymbol{M}^{p,q}_{m_{\lambda}}$ is independent of the window $\varphi$, in the sense that different (non-zero) windows in $\mathcal{S}_{\omega}(\mathbb{R}^{d})$ give equivalent norms. Indeed for $\varphi,\psi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$, $\varphi,\psi\neq 0$, we have from Proposition 2.11, applied with $\gamma=\psi$, that

[TABLE]

where $C=\frac{\|V_{\varphi}\psi\|_{L^{1}_{v_{\lambda}}}}{(2\pi)^{d}\|\psi\|_{L^{2}}^{2}}$, as we can deduce from Young inequality (3.36) (observe that $C$ is finite by Proposition 2.8 and Remark 3.4). Then, by interchanging the roles of $\varphi$ and $\psi$ we have that $V_{\varphi}f\in L^{p,q}_{m_{\lambda}}$ if and only if $V_{\psi}f\in L^{p,q}_{m_{\lambda}}$, and the corresponding modulation space norms of $f$ with respect to the two windows are equivalent.

Remark 3.6**.**

From Theorems 2.6 and 2.4 and Proposition 2.10 we have that

[TABLE]

The inversion formula of Proposition 2.10 holds also in modulation spaces, as stated in the following result.

Proposition 3.7**.**

Let $\gamma\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$ be a not identically zero window, and consider, for a measurable function $F$ on $\mathbb{R}^{2d}$, the adjoint $V_{\gamma}^{*}F$ defined as in (2.22). Then:

• (i)

The operator $V^{*}_{\gamma}$ acts continuously as

[TABLE]

and there exists $C>0$ such that

[TABLE]

where $\varphi$ is the window in the corresponding $\boldsymbol{M}^{p,q}_{m_{\lambda}}$ norm.

• (ii)

In the particular case when $F=V_{g}f$, for $g\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$, and $f\in\boldsymbol{M}^{p,q}_{m_{\lambda}}$, if $\langle\gamma,g\rangle\neq 0$ the following inversion formula holds:

[TABLE]

Proof.

(i) We start by proving that $V^{*}_{\gamma}F$ is an element of $\mathcal{S}^{\prime}_{\omega}(\mathbb{R}^{d})$. For $\psi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$ we have from (3.35),

[TABLE]

this expression is finite for $\mu>0$ sufficiently large, as we can deduce from Theorem 2.6(ii) and Definition 2.1. Then from Proposition 2.9 we have that $V^{*}_{\gamma}F$ is a well defined element of $\mathcal{S}^{\prime}_{\omega}(\mathbb{R}^{2d})$. From Theorem 2.4 we have that $V_{\varphi}V^{*}_{\gamma}F$ is a continuous function; it is explicitly given by

[TABLE]

Writing $z=(x,\xi)$ we have

[TABLE]

Then, from Young inequality (3.36) we obtain

[TABLE]

and this expression is finite since $V_{\varphi}\gamma\in\mathcal{S}_{\omega}(\mathbb{R}^{2d})\subset L^{1}_{v_{\lambda}}(\mathbb{R}^{2d})$ for every $\lambda\in\mathbb{R}$ from Remark 3.4.

(ii) We first observe that, by (3.37), $V_{g}f\in L^{p,q}_{m_{\lambda}}$. Then, from point (i), $\tilde{f}=\frac{1}{(2\pi)^{d}\langle\gamma,g\rangle}V^{*}_{\gamma}V_{g}f\in\boldsymbol{M}^{p,q}_{m_{\lambda}}$. Since $\boldsymbol{M}^{p,q}_{m_{\lambda}}\subset\mathcal{S}^{\prime}_{\omega}$, we have that $\tilde{f}=f$ by (2.24). ∎

Theorem 3.8**.**

Let $1\leq p,q<\infty$. We have

[TABLE]

and the duality is given by

[TABLE]

for $f\in\boldsymbol{M}^{p,q}_{m_{\lambda}}$ and $h\in\boldsymbol{M}^{p^{\prime},q^{\prime}}_{1/m_{\lambda}}$.

Proof.

The proof of this result relies on the duality of weighted $L^{p,q}$ spaces, and it is the same as in Theorem 11.3.6 of [G]. ∎

Proposition 3.9**.**

For $1\leq p,q<\infty$ we have that $\mathcal{S}_{\omega}(\mathbb{R}^{d})$ is a dense subspace of $\boldsymbol{M}^{p,q}_{m_{\lambda}}$.

Proof.

We first observe that, from property $(\gamma)$ of the weight function $\omega$ (see Definition 2.1) we have that, for $\mu>0$, $e^{-\mu\omega(z)}\in L^{p,q}_{m_{\lambda}}$. Hence, for every $f\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$ we obtain

[TABLE]

From Proposition 2.9 we have

[TABLE]

with continuous inclusion. It remains to prove the density. We denote by $K_{n}:=\{z\in\mathbb{R}^{2d}:|z|\leq n\}$, and we fix $\varphi\in\mathcal{S}_{\omega}$ with $\|\varphi\|_{L^{2}}^{2}=(2\pi)^{-d}$. Consider $f\in\boldsymbol{M}^{p,q}_{m_{\lambda}}$ and define

[TABLE]

From Proposition 2.5 we have that $f_{n}\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$. Moreover, using (2.24) and Proposition 3.7 we obtain

[TABLE]

So, $\|f_{n}-f\|_{\boldsymbol{M}^{p,q}_{m_{\lambda}}}$ tends to [math] for $n\to\infty$, which finishes the proof. ∎

We recall now from [G] some basic facts about amalgam spaces.

Definition 3.10**.**

We indicate with $\ell^{p,q}_{m_{\lambda}}(\mathbb{Z}^{2d})$ the space of all sequences $(a_{kn})_{k,n\in\mathbb{Z}^{d}}$, with $a_{kn}\in\mathbb{C}$ for every $k,n\in\mathbb{Z}^{d}$, such that the following norm is finite

[TABLE]

Definition 3.11**.**

Let $F$ be a measurable function on $\mathbb{R}^{2d}$, and define

[TABLE]

We say that $F\in W(L^{p,q}_{m_{\lambda}})$ if the sequence $a=(a_{kn})_{k,n\in\mathbb{Z}^{d}}$ belongs to $\ell^{p,q}_{m_{\lambda}}(\mathbb{Z}^{2d})$. The space $W(L^{p,q}_{m_{\lambda}})$ is called amalgam space, and has the norm defined by

[TABLE]

Let $\varphi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$ and $\Lambda=\alpha_{0}\mathbb{Z}^{d}\times\beta_{0}\mathbb{Z}^{d}$ a lattice with $\alpha_{0},\beta_{0}>0$ sufficiently small so that $\{\Pi(\sigma)\varphi\}_{\sigma\in\Lambda}$ is a Gabor frame for $L^{2}(\mathbb{R}^{d})$. We indicate with $\widetilde{m}_{\lambda}$ the restriction of the weight (3.34) to the lattice $\Lambda$, in the sense that

[TABLE]

We recall the following result (see Proposition 11.1.4 of [G]).

Proposition 3.12**.**

Let $F\in W(L^{p,q}_{m_{\lambda}})$ be a continuous function, and $\alpha_{0},\beta_{0}>0$. Then $F|_{\Lambda}\in\ell^{p,q}_{\widetilde{m}_{\lambda}}$, and there exists a constant $C=C(\alpha_{0},\beta_{0},\lambda)$ such that

[TABLE]

Now, we study the Gabor frame operator associated to the lattice $\Lambda$, given by

[TABLE]

for $\varphi,\psi,f\in L^{2}(\mathbb{R}^{d})$.

We write as usual $S_{\varphi,\psi}=D_{\psi}C_{\varphi}$, where $C_{\varphi}$ is the ‘analysis’ operator, acting on a function $f$ as

[TABLE]

and $D_{\psi}$ is the ‘synthesis’ operator, acting on a sequence $c=(c_{kn})_{k,n\in\mathbb{Z}^{d}}$ as

[TABLE]

We analyse the action of the previous operators on the modulation spaces $\boldsymbol{M}^{p,q}_{m_{\lambda}}$. The proofs of the next two results are very similar to [G, Thms. 12.2.3, 12.2.4], so we omit them. We just remark that, since $\varphi\in\mathcal{S}_{\omega}\subset\mathcal{S}$, we have that $V_{\varphi}\varphi\in\mathcal{S}$; then by Proposition 12.1.11 of [G] we have $V_{\varphi}\varphi\in W(L^{1}_{v_{\lambda}})$, and so we can apply Theorem 11.1.5 of [G].

Theorem 3.13**.**

Let $\varphi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$ and $\Lambda$ a lattice as before. Then the operator

[TABLE]

is bounded for every $\lambda\in\mathbb{R}\setminus\{0\}$, $\alpha_{0},\beta_{0}>0$, and $1\leq p,q\leq\infty$.

Theorem 3.14**.**

Let $\psi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$. Then we have:

• (i)

The operator

[TABLE]

is bounded, for every $1\leq p,q\leq\infty$, $\alpha_{0},\beta_{0}>0$, and $\lambda\in\mathbb{R}\setminus\{0\}$.

• (ii)

For every $c\in\ell^{p^{\prime},q^{\prime}}_{\widetilde{m}_{-\lambda}}$ and $f\in\boldsymbol{M}^{p,q}_{m_{\lambda}}$ we have that

[TABLE]

and

[TABLE]

• (iii)

For $p,q<\infty$, we have that $D_{\psi}c$ converges unconditionally in $\boldsymbol{M}^{p,q}_{m_{\lambda}}$; if $p=q=\infty$, then $D_{\psi}c$ converges unconditionally weak∗* in $\boldsymbol{M}^{\infty}_{1/v_{\lambda}}$.*

Now, we study the Gabor frame operator (3.39). We recall (see [G, Prop. 5.1.1 and 5.2.1]) that if we take a window $\varphi\in L^{2}(\mathbb{R}^{d})$ and a lattice $\Lambda$ such that $\{\Pi(\sigma)\varphi\}_{\sigma\in\Lambda}$ is a Gabor frame for $L^{2}(\mathbb{R}^{d})$, the operator (3.39) is invertible on $L^{2}(\mathbb{R}^{d})$. Moreover, if we define the dual window $\psi$ of $\varphi$ by $\psi:=S_{\varphi,\varphi}^{-1}\varphi$, we have that for every $f\in L^{2}(\mathbb{R}^{d})$,

[TABLE]

with unconditional convergence in $L^{2}(\mathbb{R}^{d})$. We observe also that if $\varphi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$ then the dual window $\psi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$ by [GZ, Thm. 4.2].

Lemma 3.15**.**

Fix $\varphi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})\setminus\{0\}$, and let $\psi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})\setminus\{0\}$ be the dual window of $\varphi$. For $f\in\boldsymbol{M}^{p,q}_{m_{\lambda}}(\mathbb{R}^{d})$, $\lambda\in\mathbb{R}\setminus\{0\}$, we have

[TABLE]

and

[TABLE]

with convergence in $\boldsymbol{M}^{p,q}_{m_{\lambda}}$ for $p,q<\infty$, and weak∗ convergence in $\boldsymbol{M}^{\infty}_{1/v_{\lambda}}$ in the case $p=q=\infty$.

Proof.

We first consider the case $p,q<\infty$. From Proposition 3.9 we have that there exists a sequence $f_{n}\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$ such that $f_{n}\to f$ in $\boldsymbol{M}^{p,q}_{m_{\lambda}}$ as $n\to\infty$. Since $\mathcal{S}_{\omega}(\mathbb{R}^{d})\subset L^{2}(\mathbb{R}^{d})$, we have that

[TABLE]

From Theorems 3.13 and 3.14 we obtain $D_{\psi}C_{\varphi}f_{n}\to D_{\psi}C_{\varphi}f$ and $D_{\varphi}C_{\psi}f_{n}\to D_{\varphi}C_{\psi}f$ in $\boldsymbol{M}^{p,q}_{m_{\lambda}}$, and so from (3.44) the result is proved.

We now pass to the case $p=q=\infty$. Let $f\in\boldsymbol{M}^{\infty}_{1/v_{\lambda}}$ and $g\in\boldsymbol{M}^{1}_{v_{\lambda}}$. We have to prove that

[TABLE]

From (3.42) and (3.43) we have that

[TABLE]

from the previous point we have that $D_{\varphi}C_{\psi}g=g$ in $\boldsymbol{M}^{1}_{v_{\lambda}}$, so the first equality in (3.45) is proved. The other is similar. ∎

Remark 3.16**.**

Let $u\in\mathcal{S}_{\omega}^{\prime}(\mathbb{R}^{d})$, and $\varphi,\psi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$ as in Lemma 3.15. Then for every $\theta\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$ we have

[TABLE]

We have indeed that from Remark 3.6 there exists $\lambda<0$ such that $u\in\boldsymbol{M}^{\infty}_{m_{\lambda}}=\boldsymbol{M}^{\infty}_{1/v_{\lambda}}$. Then, from Lemma 3.15, for every $g\in\boldsymbol{M}^{1}_{v_{\lambda}}$,

[TABLE]

From Proposition 3.9, the previous formula then holds for $g=\theta\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$, so we have (3.46). **

We can now prove the main result of this section.

Theorem 3.17**.**

If $u\in\mathcal{S}^{\prime}_{\omega}(\mathbb{R}^{d})$ then

[TABLE]

Proof.

The inclusion $\mathop{\rm WF}\nolimits^{G}_{\omega}(u)\subseteq\mathop{\rm WF}\nolimits^{\prime}_{\omega}(u)$ is trivial, so that we only have to prove that

[TABLE]

Let $0\neq z_{0}\notin\mathop{\rm WF}\nolimits^{G}_{\omega}(u)$. So, there exists an open conic set $\Gamma\subset\mathbb{R}^{2d}\setminus\{0\}$ containing $z_{0}$ such that (3.33) is satisfied. By Remark 3.16 we have that, for $\varphi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})\setminus\{0\}$ and $\tilde{\varphi}=S_{\varphi\varphi}^{-1}\varphi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$ its dual window,

[TABLE]

We denote

[TABLE]

Clearly $V_{\varphi}u(z)=V_{\varphi}u_{1}(z)+V_{\varphi}u_{2}(z)$. Denoting $\sigma=(\sigma_{1},\sigma_{2})\in\mathbb{R}^{d}\times\mathbb{R}^{d}$, by (2.10), (2.11), the subadditivity of $\omega$ and (2.13), we can estimate, for every $\alpha,\beta\in\mathbb{N}_{0}^{d}$, $\lambda,\mu>0$:

[TABLE]

for some $C_{\lambda},C_{\lambda,\lambda^{\prime},\mu}>0$.

For $\lambda^{\prime}=6\lambda$ we apply [BJ, Prop. 2.1(g)], then (2.10) and (3.33), and finally obtain, for some constants depending on $\lambda$ and $\mu$:

[TABLE]

This proves that $u_{1}\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$ (here, we consider the seminorms given in (2.1)). Therefore, from Theorem 2.6, $V_{\varphi}u_{1}\in\mathcal{S}_{\omega}(\mathbb{R}^{2d})$ and for every $\lambda>0$ there is a constant $C_{\lambda}>0$ such that

[TABLE]

Let us now fix an open conic set $\Gamma^{\prime}\subset\mathbb{R}^{2d}\setminus\{0\}$ containing $z_{0}$ and such that $\overline{\Gamma^{\prime}\cap S_{2d-1}}\subseteq\Gamma$.

Then

[TABLE]

and $|\sigma-z|\geq\varepsilon|\sigma|$ for $0\neq\sigma\in\Lambda\setminus\Gamma$ and $z\in\Gamma^{\prime}$.

From the subadditivity of $\omega$ we have

[TABLE]

for some $C,\bar{\lambda}>0$, because of Theorem 2.4 and since ([G, pg 41])

[TABLE]

Since $\tilde{\varphi}\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$, from Theorem 2.6 we have that for every $\mu>0$ there is a constant $C_{\mu}>0$ such that

[TABLE]

and hence, substituting in (3.50):

[TABLE]

However, for $z\in\Gamma^{\prime}$ and $\sigma\in\Lambda\setminus\Gamma$ we have $|\sigma-z|\geq\varepsilon|\sigma|$ and therefore, by the subadditivity of $\omega$,

[TABLE]

for some $M>0$ depending on the constant $\varepsilon$ defined in (3.49). By formula (3.52) we obtain

[TABLE]

for some $C_{\lambda}>0$, if $\mu$ is chosen large enough.

From (3.48) and (3.53) we finally deduce

[TABLE]

and hence $z_{0}\notin\mathop{\rm WF}\nolimits^{\prime}_{\omega}(u)$. ∎

From Theorem 3.17, in what follows we use $\mathop{\rm WF}\nolimits^{\prime}_{\omega}(u)$ for $\mathop{\rm WF}\nolimits^{G}_{\omega}(u)$ and any $u\in\mathcal{S}^{\prime}_{\omega}(\mathbb{R}^{d})$.

Proposition 3.18**.**

For every $u\in\mathcal{S}^{\prime}_{\omega}(\mathbb{R}^{d})$ we have $\mathop{\rm WF}\nolimits^{\prime}_{\omega}(u)=\emptyset$ if and only if $u\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$.

Proof.

Suppose that $u\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$, and fix a window function $\varphi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})\setminus\{0\}$; then from Theorem 2.6 we have that for every $\lambda>0$ there exists $C_{\lambda}>0$ such that

[TABLE]

Then for every open conic set $\Gamma\subseteq\mathbb{R}^{2d}\setminus\{0\}$ condition (3.25) holds, so $\mathop{\rm WF}\nolimits^{\prime}_{\omega}(u)=\emptyset$. Suppose now that $\mathop{\rm WF}\nolimits^{\prime}_{\omega}(u)=\emptyset$. From Definition 3.1 we have that for every $\overline{z}\in\mathbb{R}^{2d}\setminus\{0\}$ there exists an open conic set $\Gamma_{\overline{z}}\subseteq\mathbb{R}^{2d}\setminus\{0\}$ containing $\overline{z}$ such that for every $\lambda>0$ there exists $C_{\lambda,\overline{z}}>0$ satisfying

[TABLE]

Let $\Upsilon_{\overline{z}}=\Gamma_{\overline{z}}\cap S_{2d-1}$. We have that $\{\Upsilon_{\overline{z}},\ \overline{z}\in\mathbb{R}^{2d}\setminus\{0\}\}$ is an open covering of $S_{2d-1}$; since $S_{2d-1}$ is compact and $\Gamma_{\overline{z}}$ is conic, there exist $z_{1},\dots,z_{k}\in\mathbb{R}^{2d}\setminus\{0\}$ such that

[TABLE]

We then have that for every $\lambda>0$,

[TABLE]

where $C_{\lambda}=\max\{C_{\lambda,z_{1}},\dots,C_{\lambda,z_{k}},|V_{\varphi}u(0)|e^{\lambda\omega(0)}\}$. From Theorem 2.6 we finally have $u\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$. ∎

We now prove that the wave front set $\mathop{\rm WF}\nolimits^{\prime}_{\omega}$ is not affected by the phase-space shift operator.

Proposition 3.19**.**

For every $w=(y,\eta)\in\mathbb{R}^{2d}$ and for every $u\in\mathcal{S}^{\prime}_{\omega}(\mathbb{R}^{d})$ we have

[TABLE]

Proof.

Since $\Pi(w)=M_{\eta}T_{y}$, it is enough to prove that translation and modulation do not affect the wave front set. Concerning translation, we have that for $z=(x,\xi)\in\mathbb{R}^{2d}$,

[TABLE]

writing $\psi=T_{-y}\varphi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$ we have that

[TABLE]

and since the wave front set does not depend on the window (Proposition 3.2) we have $\mathop{\rm WF}\nolimits^{\prime}_{\omega}(T_{y}u)=\mathop{\rm WF}\nolimits^{\prime}_{\omega}(u)$. Concerning modulation, we have

[TABLE]

then, writing $\theta=M_{-\eta}\varphi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$, we get

[TABLE]

and as before we conclude that $\mathop{\rm WF}\nolimits^{\prime}_{\omega}(M_{\eta}u)=\mathop{\rm WF}\nolimits^{\prime}_{\omega}(u)$. ∎

The results obtained in Sections 2 and 3 are true with the weaker assumption (see Björck [B]): “there exist $a\in\mathbb{R},b>0$ such that $\omega(t)\geq a+b\log(1+t)$ for $t\geq 0$” instead of $(\gamma)$ of Definition 2.1. A detailed and modern treatment of these type of weights can be found [BG]. Moreover, the results above are true in the quasi-analytic case also, i.e. when we consider that $\omega(t)=o(t)$, as $t\to+\infty$, instead of condition $(\beta)$ of Definition 2.1.

4. Applications to (pseudo-)differential operators

In this section we analyze the action of several operators of pseudo-differential (or differential) type on the global wave front set $\mathop{\rm WF}\nolimits^{\prime}_{\omega}(u)$ of $u\in\mathcal{S}^{\prime}_{\omega}(\mathbb{R}^{d})$. We will use the kernel theorem in $\mathcal{S}_{\omega}$. It is known that $\mathcal{S}_{\omega}$ is nuclear for many weight functions $\omega$. For example, whenever they satisfy the following condition:

[TABLE]

Morever, Bonet, Meise and Melikhov [BMM] proved that under such a condition the classes of ultradifferentiable functions defined by sequences in the sense of Komatsu satisfying the standard conditions $(M0)$, $(M1)$, $(M2)$ and $(M3)$, and the classes defined by weight functions in the sense of Braun, Meise and Taylor [BMT] coincide. Hence, under condition (4.54) our results are true also for spaces of the type we are treating defined by sequences (see, for instance, Langenbruch [L] for the definition of the spaces and several properties of them).

We start by defining the following symbol class.

Definition 4.1**.**

For $m\in\mathbb{R}$ we define

[TABLE]

Then we consider the Kohn-Nirenberg quantization defined by

[TABLE]

The above Kohn-Nirenberg quantization is well defined since $\widehat{f}\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$ and hence for every $\lambda>0$ there exists $C_{\lambda}>0$ such that

[TABLE]

which is integrable in $\mathbb{R}^{d}$ if we choose $\lambda>0$ sufficiently large. Moreover,

[TABLE]

If $\mathcal{S}_{\omega}$ is nuclear, we can use the kernel theorem and find a unique distribution $K\in\mathcal{S}^{\prime}_{\omega}(\mathbb{R}^{4d})$ of the linear operator

[TABLE]

such that

[TABLE]

in the sense that

[TABLE]

If $u\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$ and $F=V_{\varphi}u\in\mathcal{S}_{\omega}(\mathbb{R}^{2d})$ for $\varphi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$ with $\|\varphi\|_{L^{2}}=1$, then, from (2.21),

[TABLE]

and we can compute the kernel directly:

Lemma 4.2**.**

For $a\in S^{m}_{\omega}$, $\varphi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$ with $\|\varphi\|_{L^{2}}=1$ and $u\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$ we have that

[TABLE]

where, for all $z=(y,\eta),z^{\prime}=(y^{\prime},\eta^{\prime})\in\mathbb{R}^{2d}$,

[TABLE]

Proof.

Let $F\in\mathcal{S}_{\omega}(\mathbb{R}^{2d})$ and consider the Kohn-Nirenberg quantization (4.55) of $V_{\varphi}^{*}F\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$:

[TABLE]

Then, by (2.22),

[TABLE]

Since $a\in S^{m}_{\omega}$, $F\in\mathcal{S}_{\omega}(\mathbb{R}^{2d})$ and $\varphi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$, we have that for every $\lambda_{1},\lambda_{2},\lambda_{3}>0$ there exists a constant $C_{\lambda}>0$ such that

[TABLE]

Choosing $\lambda_{1},\lambda_{2}>0$ sufficiently large we can apply Fubini’s theorem with respect to the variables $y,\eta$ and $x^{\prime}$, obtaining:

[TABLE]

Since $a\in S^{m}_{\omega}$, $F\in\mathcal{S}_{\omega}(\mathbb{R}^{2d})$ and $\varphi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$, for every $\mu_{1},\mu_{2},\mu_{3}>0$ there exists a constant $C_{\mu}>0$ such that

[TABLE]

so that, for $\mu_{3},\mu_{1}>\mu_{2}$ sufficiently large, we can apply Fubini’s theorem and obtain

[TABLE]

Applying the above result to $F=V_{\varphi}u$ for some $u\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$, since $\|\varphi\|_{L^{2}}=1$ and hence $V_{\varphi}^{*}F=V_{\varphi}^{*}V_{\varphi}u=(2\pi)^{d}u$ by (2.21), we have

[TABLE]

for

[TABLE]

which concludes the proof of the lemma. ∎

In the next result the following property on the weight function $\omega$ will be useful: from [BJO, Lemma 4.7(ii)] (for instance), for every $\mu>0$ and $t\geq 1$,

[TABLE]

where $a\in\mathbb{R}$ and $b>0$ are constants that depend on $\omega$.

Proposition 4.3**.**

If $a\in S^{m}_{\omega}$, $m\in\mathbb{R}$ and $K\in C^{\infty}(\mathbb{R}^{4d})$ is defined by (4.59), then for every $\lambda>0$ there exists a constant $C_{\lambda}>0$ such that

[TABLE]

Moreover, if $a(z)=0$ for $z\in\Gamma\setminus\overline{B(0,R)}$ for an open conic set $\Gamma\subseteq\mathbb{R}^{2d}\setminus\{0\}$ and for some $R>0$ (here $B(0,R)$ is the ball of center [math] and radius $R$ in $\mathbb{R}^{2d}$), then for every open conic set $\Gamma^{\prime}\subseteq\mathbb{R}^{2d}\setminus\{0\}$ such that $\overline{\Gamma^{\prime}\cap S_{2d-1}}\subseteq\Gamma$ we have that for every $\lambda>0$ there exists a constant $C_{\lambda}>0$ such that for all $z^{\prime}=(y^{\prime},\eta^{\prime})\in\Gamma^{\prime}$ and $z=(y,\eta)\in\mathbb{R}^{2d}$,

[TABLE]

Proof.

By the linear change of variables $\xi^{\prime}=\xi-\eta$ and $x^{\prime}=x-y^{\prime}$ in (4.59) we have

[TABLE]

and hence, setting $x=x^{\prime}$ and $\xi=\xi^{\prime}$:

[TABLE]

Writing, for $M,N\in\mathbb{N}_{0}$,

[TABLE]

and integrating by parts in (4.63), we have

[TABLE]

where

[TABLE]

For $a\in S^{m}_{\omega}$, since $\varphi,\widehat{\varphi}\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$, by [BJO, Thm. 4.8(5)] we have that for every $\lambda$, $\mu$, $\lambda^{\prime}$, $\mu^{\prime}$, $\lambda^{\prime\prime}$, $\mu^{\prime\prime}>0$ there are positive constants $C_{\lambda,\mu},C_{\lambda^{\prime},\lambda^{\prime\prime}},C_{\mu^{\prime},\mu^{\prime\prime}}$, depending only on the indexed constants, such that for every $M,N,k,\ell\in\mathbb{N}_{0}$:

[TABLE]

Note that

[TABLE]

Moreover, we can choose $\lambda^{\prime\prime}=\lambda$, $\mu^{\prime\prime}=\mu$ and apply Proposition 2.1(g) of [BJ]. Taking also into account the subadditivity of $\omega$, we have that for every $\lambda$, $\mu$, $\lambda^{\prime}$, $\mu^{\prime}>0$ there is a constant $C_{\lambda,\mu,\lambda^{\prime},\mu^{\prime}}>0$ such that for all $M,N,k,\ell\in\mathbb{N}_{0}$:

[TABLE]

Taking the infimum on $k\in\mathbb{N}_{0}$ and applying (4.60) and (2.11), we get:

[TABLE]

Substituting in (4.64) we have that for all $\lambda,\mu,\lambda^{\prime},\mu^{\prime}>0$ there is a constant $C^{\prime}_{\lambda,\mu,\lambda^{\prime},\mu^{\prime}}>0$ such that for every $M,N,\ell\in\mathbb{N}_{0}$:

[TABLE]

Let us now fix $\mu^{\prime}>0$, choose $\ell\in\mathbb{N}$ and $\lambda^{\prime}>0$ sufficiently large so that the above integrals converge, take the infimum on $M$ and $N$, apply (4.60) and obtain:

[TABLE]

In particular, by the arbitrariness of $\lambda$ and $\mu$ in (4.68), we have that for every $\lambda,\mu>0$ there is a constant $C_{\lambda,\mu}>0$ such that

[TABLE]

which proves (4.61) for $\mu=\lambda$.

Applying (4.66) only to $\langle\eta-\eta^{\prime}+\xi\rangle^{-M}$ in (4.65), by the same computations as to get (4.67) we obtain that if $a(z)=0$ for $z\in\Gamma\setminus\overline{B(0,R)}$, then

[TABLE]

where

[TABLE]

We now want to estimate (4.70) for $z^{\prime}=(y^{\prime},\eta^{\prime})\in\Gamma^{\prime}$ and $z=(y,\eta)\in\mathbb{R}^{2d}$. It has been proved in [RW, pg 643] that

[TABLE]

for some constant $C>0$.

Substituting (4.71) into (4.70) and applying [BJ, Prop. 2.1(g)] we have, for $z^{\prime}\in\Gamma^{\prime}\setminus B(0,2R)$ and $z\in\mathbb{R}^{2d}$:

[TABLE]

We now fix $\mu^{\prime}>0$ and choose $\ell\in\mathbb{N}$ and $\lambda^{\prime}>0$ sufficiently large so that the above integral is convergent; then take the infimum on $M$ and $N$ and apply (4.60). We obtain that for every $\lambda,\mu>0$ there is a constant $C_{\lambda,\mu}>0$ such that

[TABLE]

In particular, for $\bar{\lambda}=\frac{\lambda}{6}-\frac{1}{b}$ and $\bar{\mu}=\frac{\mu}{12}-\frac{2}{b}$ we have that there is $C_{\bar{\lambda},\bar{\mu}}>0$ such that, for $z^{\prime}\in\Gamma^{\prime}\setminus B(0,2R)$ and $z\in\mathbb{R}^{2d}$:

[TABLE]

For $\bar{\mu}\geq 4\bar{\lambda}+2m$ we have that $m-\bar{\mu}\leq-\bar{\lambda}$ and $-\frac{\bar{\mu}}{2}\leq m-\frac{\bar{\mu}}{2}\leq-2\bar{\lambda}$ which proves (4.62) for $z^{\prime}\in\Gamma^{\prime}\setminus B(0,2R),z\in\mathbb{R}^{2d}$.

Since the estimate (4.62) for $|z^{\prime}|\leq 2R$ follows from (4.69), the proof is complete. ∎

Remark 4.4**.**

For $a\in S^{m}_{\omega}$, $m\in\mathbb{R},$ and $K\in C^{\infty}(\mathbb{R}^{4d})$ defined by (4.59) the integral in (4.58) is well defined also for $u\in\mathcal{S}^{\prime}_{\omega}(\mathbb{R}^{d})$. In fact, (2.7) and (4.61) imply that there exist $\tilde{C},\tilde{\lambda}>0$ and that for every $\lambda>0$ there exists $C_{\lambda}>0$ such that

[TABLE]

if $\lambda>\max\{\tilde{\lambda},m+\tilde{\lambda}\}$. **

We now want to extend Lemma 4.2 for $u\in\mathcal{S}^{\prime}_{\omega}(\mathbb{R}^{d})$. To this aim we first need the next two results.

Proposition 4.5**.**

The space $\mathcal{S}_{\omega}(\mathbb{R}^{d})$ is dense in $\mathcal{S}^{\prime}_{\omega}(\mathbb{R}^{d})$.

Proof.

Let us consider the inclusion

[TABLE]

To show that the image is dense we take $T\in\left(\mathcal{S}^{\prime}_{\omega}(\mathbb{R}^{d})\right)^{\prime}$ such that $\left.T\right|_{\mathcal{S}_{\omega}}=0$ and prove that $T\equiv 0$.

Since $\mathcal{S}_{\omega}(\mathbb{R}^{d})$ is reflexive, there exists a unique $f\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$ such that

[TABLE]

because of $\left.T\right|_{\mathcal{S}_{\omega}}=0$. Therefore $f=0$, i.e. $T\equiv 0$. ∎

Proposition 4.6**.**

Let $\varphi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})\setminus\{0\}$. Then

[TABLE]

is continuous.

Proof.

We already know that

[TABLE]

is continuous by (2.20). It follows that

[TABLE]

is continuous and moreover $\left.(V_{\varphi}^{*})^{*}\right|_{\mathcal{S}_{\omega}(\mathbb{R}^{d})}=V_{\varphi}$ because, for $f,g\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$,

[TABLE]

Since $\mathcal{S}_{\omega}(\mathbb{R}^{d})$ is dense in $\mathcal{S}^{\prime}_{\omega}(\mathbb{R}^{d})$ by Proposition 4.5, we have that $(V_{\varphi}^{*})^{*}$ is the continuous extension of $V_{\varphi}$ to $\mathcal{S}^{\prime}_{\omega}(\mathbb{R}^{d})$ and, hence, $V_{\varphi}$ is continuous on $\mathcal{S}^{\prime}_{\omega}(\mathbb{R}^{d})$ also. ∎

Now, we need amplitudes $a(x,y,\xi)$, instead of symbols $a(x,\xi)$.

Definition 4.7**.**

Given $m\in\mathbb{R}$, we say that $a(x,y,\xi)\in C^{\infty}(\mathbb{R}^{3d})$ is an amplitude in the space $S^{m}_{\omega}$ if for every $\lambda,\mu>0$ there is $C_{\lambda,\mu}>0$ such that

[TABLE]

for all $(x,y,\xi)\in\mathbb{R}^{3d}$ and $\alpha,\beta,\gamma\in\mathbb{N}_{0}^{d}$.

Now, proceeding in a similar way to that of Proposition 1.9 and Theorem 2.2 of [FGJ], one can prove that if $a(x,y,\xi)\in S^{m}_{\omega}$ is an amplitude as in Definition 4.7, the operator acting on $\mathcal{S}_{\omega}$, given by the iterated integral

[TABLE]

is well defined and continuous from $\mathcal{S}_{\omega}$ into itself. The operator $A$ is called pseudo-differential operator of type $\omega$ with amplitude $a(x,y,\xi)$. Moreover, $A$ can be extended continuously to the dual space $\tilde{A}:\mathcal{S}^{\prime}_{\omega}\to\mathcal{S}^{\prime}_{\omega}$ in a standard way (see [FGJ, Theorem 2.5]). In particular, the Kohn-Nirenberg quantization defined in (4.55) is a pseudo-differential operator with amplitude

[TABLE]

where $p(x,\xi)$ is a symbol as in Definition 4.1.

As a consequence of the above considerations and of the estimates of the kernel in Proposition 4.3, we obtain the following result:

Corollary 4.8**.**

Let $a(x,\xi)\in S^{m}_{\omega}$ a symbol as in Definition 4.1, $\varphi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$ with $\|\varphi\|_{L^{2}}=1$ and $u\in\mathcal{S}^{\prime}_{\omega}(\mathbb{R}^{d})$. Then, for $K(z^{\prime},z)$ as in (4.59), we have

[TABLE]

for all $z^{\prime}\in\mathbb{R}^{2d}$.

Proof.

Since $V_{\varphi}$ operates on $\mathcal{S}^{\prime}_{\omega}$, from the previous comments it is clear that $V_{\varphi}a(x,D)$ can be extended to $\mathcal{S}^{\prime}_{\omega}(\mathbb{R}^{d})$. We take $u\in\mathcal{S}^{\prime}_{\omega}(\mathbb{R}^{d})$. By Proposition 4.5, there exists a sequence $\{u_{n}\}_{n\in\mathbb{N}}\subset\mathcal{S}_{\omega}(\mathbb{R}^{d})$ which converges to $u$ in $\mathcal{S}^{\prime}_{\omega}$ and, hence,

[TABLE]

We want to prove that

[TABLE]

using Lebesgue’s dominated convergence theorem. First, it is easy to see that $\{V_{\varphi}u_{n}(z)\}_{n\in\mathbb{N}}$ converges pointwise to $V_{\varphi}u(z)$ for every $z\in\mathbb{R}^{2d}$ from the definition of the short-time Fourier transform.

Now, since $\{u_{n}\}_{n\in\mathbb{N}}$ is bounded in $\mathcal{S}^{\prime}_{\omega}(\mathbb{R}^{d})$, it is equicontinuous there. So, there exist a constant $C>0$ and a seminorm $q$ on $\mathcal{S}_{\omega}(\mathbb{R}^{d})$ such that

[TABLE]

This yields a uniform estimate of the inequality (2.7) (see the proof of [GZ, Theorem 2.4]) in the sense:

[TABLE]

for some $\tilde{C},\tilde{\lambda}>0$ independent of $n$ and $z$. From (4.76) and (4.72) we have that $K(z^{\prime},z)V_{\varphi}u_{n}(z)$ is dominated by a function in $L^{1}(\mathbb{R}^{2d}_{z})$.

Therefore (4.75) is satisfied and hence, from (4.74),

[TABLE]

also for $u\in\mathcal{S}^{\prime}_{\omega}(\mathbb{R}^{d})$. ∎

We recall the notion of conic support from [RW]:

Definition 4.9**.**

For $a\in\mathcal{D}^{\prime}(\mathbb{R}^{2d})$ the conic support of $a$, denoted by $\mathop{\rm cone\,supp}(a)$, is the set of all $z\in\mathbb{R}^{2d}\setminus\{0\}$ such that any open conic set $\Gamma\subset\mathbb{R}^{2d}\setminus\{0\}$ containing $z$ satisfies that

[TABLE]

We have the following

Proposition 4.10**.**

If $m\in\mathbb{R}$, $a\in S^{m}_{\omega}$ and $u\in\mathcal{S}^{\prime}_{\omega}(\mathbb{R}^{d})$, then

[TABLE]

Proof.

Let $0\neq z_{0}\notin\mathop{\rm cone\,supp}(a)$. This means that there exists an open conic set $\Gamma\subset\mathbb{R}^{2d}\setminus\{0\}$ containing $z_{0}$ and such that $a(z)=0$ for $z\in\Gamma\setminus\overline{B(0,R)}$ for some $R>0$. Then, from Proposition 4.3, for every open conic set $\Gamma^{\prime}\subseteq\mathbb{R}^{2d}\setminus\{0\}$ with $\overline{\Gamma^{\prime}\cap S_{2d-1}}\subseteq\Gamma$ we have that the kernel $K(z^{\prime},z)$ defined by (4.59) satisfies the estimate (4.62) for all $z^{\prime}\in\Gamma^{\prime}$ and $z\in\mathbb{R}^{2d}$.

We argue as in Corollary 4.8 and use (4.62) to obtain that formula (4.73) holds for all $z^{\prime}\in\Gamma^{\prime}$ and therefore there exist $C,\bar{\lambda}>0$ and for every $\lambda,N>0$ there exists $C_{\lambda,N}>0$ such that, for all $z^{\prime}\in\Gamma^{\prime}$,

[TABLE]

It follows, by the subadditivity of $\omega$, that

[TABLE]

for some $C_{\lambda}>0$ if we choose $N$ sufficiently large so that the integral in (4.77) converges.

This proves that $z_{0}\notin\mathop{\rm WF}\nolimits^{\prime}_{\omega}(a(x,D)u)$ by Definition 3.1, and the proof is complete. ∎

Since our weight functions are non-quasianalytic, we can obtain the following consequence of Proposition 4.10.

Corollary 4.11**.**

Let $a\in\mathcal{S}_{\omega}(\mathbb{R}^{2d})$ with compact support, and consider the corresponding pseudo-differential operator $a(x,D)$, cf. (4.55). Then $a(x,D)$ is globally $\omega$-regularizing, in the sense that for every $u\in\mathcal{S}^{\prime}_{\omega}(\mathbb{R}^{d})$ we have $a(x,D)u\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$.

Proof.

It is easy to see that $a\in S^{0}_{\omega}$. Consequently, the corresponding pseudo-differential operator $a(x,D)$ can be extended to $\mathcal{S}^{\prime}_{\omega}(\mathbb{R}^{d})$. Since the support of $a$ is compact, we have that $\mathop{\rm cone\,supp}(a)=\emptyset$. From Proposition 4.10 we get $\mathop{\rm WF}\nolimits^{\prime}_{\omega}(a(x,D)u)=\emptyset.$ We apply Proposition 3.18 to conclude. ∎

In the next part of the section we consider other kind of operators, proving that they do not enlarge the wave front set. We start from the operators with polynomial coefficients.

Theorem 4.12**.**

Let $m>0$ be an integer, and consider

[TABLE]

where $c_{\alpha\beta}\in\mathbb{C}$. Then for every $u\in\mathcal{S}_{\omega}^{\prime}(\mathbb{R}^{d})$ we have

[TABLE]

Proof.

We fix a window function $\varphi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$, and, for $\nu\in\mathbb{N}^{d}_{0}$ we write $\varphi_{\nu}$ for the function

[TABLE]

For every $\alpha\in\mathbb{N}^{d}_{0}$ and $z=(y,\eta)\in\mathbb{R}^{2d}$ we obtain by induction on $|\alpha|$ that

[TABLE]

We have indeed that for $|\alpha|=1$, writing $\boldsymbol{1}_{j}$ for the multi-index in $\mathbb{N}^{d}_{0}$ having $1$ in the $j$-th position and [math] elsewhere, we have

[TABLE]

we suppose now that (4.78) is true for every $|\alpha|=n$, and prove it for $\tilde{\alpha}$ with $|\tilde{\alpha}|=n+1$. There exists $j\in\{1,\dots,d\}$ such that $\tilde{\alpha}=\alpha+\boldsymbol{1}_{j}$. Then by the inductive hypothesis we have

[TABLE]

and so (4.78) is proved. From the definition of short-time Fourier transform we have

[TABLE]

and so by (4.78) we get

[TABLE]

Concerning derivation, since

[TABLE]

a direct computation shows that

[TABLE]

From (4.79) and (4.80) we finally obtain

[TABLE]

On the other hand, it is not difficult to see that for every $\mu,\nu\in\mathbb{N}^{d}_{0}$, $D^{\mu}\varphi_{\nu}\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$.

Suppose now that $z_{0}=(y_{0},\eta_{0})\notin\mathop{\rm WF}\nolimits^{\prime}_{\omega}(u)$, $z_{0}\in\mathbb{R}^{2d}\setminus\{0\}$. Then, there exists an open conic set $\Gamma\subseteq\mathbb{R}^{2d}\setminus\{0\}$ containing $z_{0}$ and such that

[TABLE]

From Proposition 3.2 we have that for every $\mu,\nu\in\mathbb{N}^{d}_{0}$ and for every open conic set $\Gamma^{\prime}\subseteq\mathbb{R}^{2d}\setminus\{0\}$ containing $z_{0}$ and such that $\overline{\Gamma^{\prime}\cap S_{2d-1}}\subseteq\Gamma$,

[TABLE]

From (4.81), for every $k>0$ we get

[TABLE]

Since $|\alpha-\nu|+|\beta-\mu|\leq m$, from the property $(\gamma)$ of the weight function $\omega$ we obtain

[TABLE]

for every $\nu\leq\alpha$, $\mu\leq\beta$. Therefore, from (4.82) we obtain

[TABLE]

which means that $z_{0}\notin\mathop{\rm WF}\nolimits^{\prime}_{\omega}(A(x,D)u)$, and the proof is complete. ∎

We now want to prove an analogue of Theorem 4.12 for the case of localization operators. We recall here the definition of such operators and prove some results that are needed for our purpose. Given two window functions $\psi,\gamma\in\mathcal{S}_{\omega}(\mathbb{R}^{d})\setminus\{0\}$ and a symbol $a\in\mathcal{S}^{\prime}_{\omega}(\mathbb{R}^{2d})$, the corresponding localization operator $L^{a}_{\psi,\gamma}$ is defined, for $f\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$, as

[TABLE]

From Proposition 2.8 we have that

[TABLE]

We want now to consider symbols in a smaller class than $\mathcal{S}^{\prime}_{\omega}(\mathbb{R}^{2d})$, in order to apply the corresponding localization operator to distributions. We have the following result.

Lemma 4.13**.**

Let $a(z)$, $z\in\mathbb{R}^{2d}$, be a measurable function such that there exist $\tau,C>0$ such that

[TABLE]

Then

[TABLE]

and

[TABLE]

are continuous.

Proof.

Let $f\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$. From Theorem 2.6 we have that for every $\lambda,\rho>0$ there exists $C_{\lambda}>0$ such that

[TABLE]

and so, choosing $\lambda\geq\rho+\tau$, we have that $a\cdot V_{\psi}f\in L^{\infty}_{m_{\rho}}(\mathbb{R}^{2d})$ for every $\rho>0$, where $m_{\rho}$ is defined by (3.34). From Proposition 3.7 and (4.83), we have that $L^{a}_{\psi,\gamma}f\in\boldsymbol{M}^{\infty}_{m_{\rho}}(\mathbb{R}^{d})$ for every $\rho>0$, and then, from Remark 3.6, $L^{a}_{\psi,\gamma}f\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$. To prove the continuity of $L^{a}_{\psi,\gamma}$ on $\mathcal{S}_{\omega}(\mathbb{R}^{d})$ we fix $\varphi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})\setminus\{0\}$, $\rho>0$, and we observe that from (3.38) (with $p=q=\infty$) and (4.84) we get

[TABLE]

From Proposition 2.9 we have that (4.85) is continuous. Let now $f\in\mathcal{S}_{\omega}^{\prime}(\mathbb{R}^{d})$. From Remark 3.6 there exists $\lambda<0$ such that $f\in\boldsymbol{M}^{\infty}_{m_{\lambda}}(\mathbb{R}^{d})$; then, choosing $\rho=-|\tau|-|\lambda|$ we have

[TABLE]

for every $z\in\mathbb{R}^{2d}$, so $a\cdot V_{\psi}f\in L^{\infty}_{m_{\rho}}(\mathbb{R}^{2d})$. Then by Proposition 3.7 we have $L^{a}_{\psi,\gamma}f\in\boldsymbol{M}^{\infty}_{m_{\rho}}(\mathbb{R}^{d})$, and from Remark 3.6 we finally have $L^{a}_{\psi,\gamma}f\in\mathcal{S}^{\prime}_{\omega}(\mathbb{R}^{d})$. Observe now that for every $u\in\mathcal{S}_{\omega}^{\prime}(\mathbb{R}^{d})$ and $v\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$ we have

[TABLE]

Then $L^{a}_{\psi,\gamma}=(L^{\overline{a}}_{\gamma,\psi})^{*}$; since $\overline{a}$ satisfies the same estimates as $a$, the continuity of (4.86) follows from that of (4.85). ∎

Theorem 4.14**.**

Let $\psi,\gamma\in\mathcal{S}_{\omega}(\mathbb{R}^{d})\setminus\{0\}$, and let $a$ be a symbol satisfying (4.84). Then for every $u\in\mathcal{S}^{\prime}_{\omega}(\mathbb{R}^{d})$ we have

[TABLE]

Proof.

Let $z_{0}\notin\mathop{\rm WF}\nolimits^{\prime}_{\omega}(u)$, $z_{0}\in\mathbb{R}^{2d}\setminus\{0\}$. Then there exists an open conic set $\Gamma\subseteq\mathbb{R}^{2d}\setminus\{0\}$ containing $z_{0}$ such that

[TABLE]

From (4.84), since $\lambda$ is arbitrary we have

[TABLE]

For windows functions $\varphi,\gamma\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$ we can then repeat the same procedure used in the proof of Proposition 3.2. First, we observe that from de definition of localization operator

[TABLE]

Now, it is not difficult to see that

[TABLE]

and hence

[TABLE]

Consequently, for every open conic set $\Gamma^{\prime}\subseteq\mathbb{R}^{2d}\setminus\{0\}$ containing $z_{0}$ and such that $\overline{\Gamma^{\prime}\cap S_{2d-1}}\subseteq\Gamma$ we have (see the proof of Proposition 3.2)

[TABLE]

This implies that $z_{0}\notin\mathop{\rm WF}\nolimits^{\prime}_{\omega}(L^{a}_{\psi,\gamma}u)$ and the proof is complete. ∎

5. Examples

In this section we compute the Gabor wave front set for some particular $u\in\mathcal{S}^{\prime}_{\omega}(\mathbb{R}^{d})$ (see also the examples in [RW]).

Example 5.1**.**

Consider the Dirac distribution $u=\delta\in\mathcal{S}_{\omega}^{\prime}(\mathbb{R}^{d})$ for every weight $\omega$. We have that

[TABLE]

Since $V_{\varphi}\delta(0,\xi)=\overline{\varphi(0)}$, choosing $\varphi$ in such a way that $\varphi(0)\neq 0$ we have

[TABLE]

Let now $(x_{0},\xi_{0})\in\mathbb{R}^{2d}\setminus\{0\}$ such that $x_{0}\neq 0$, and consider an open conic set containing $(x_{0},\xi_{0})$ of the form

[TABLE]

for $C>0$. From the subadditivity of $\omega$, there exists $C_{1}>0$ such that, writing $z=(x,\xi)$,

[TABLE]

since $\varphi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$. Then $(x_{0},\xi_{0})\notin\mathop{\rm WF}\nolimits^{\prime}_{\omega}(\delta)$, and so $\mathop{\rm WF}\nolimits^{\prime}_{\omega}(\delta)=\{0\}\times(\mathbb{R}^{d}\setminus\{0\})$. From Proposition 3.19 we have that for every $\overline{x}\in\mathbb{R}^{d}$, writing $\delta_{\overline{x}}$ for the Dirac distribution centered at $\overline{x}$,

[TABLE]

Example 5.2**.**

Let $u=\boldsymbol{1}$ be the function identically $1$, that belong to $\mathcal{S}^{\prime}_{\omega}(\mathbb{R}^{d})$ for every weight $\omega$. A direct computation shows that

[TABLE]

since $\hat{\varphi}\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$ we can proceed as in Example 5.1, obtaining that for every weight $\omega$, $\mathop{\rm WF}\nolimits^{\prime}_{\omega}(\boldsymbol{1})=(\mathbb{R}^{d}\setminus\{0\})\times\{0\}$. From Proposition 3.19 we then have that for every $\overline{\xi}\in\mathbb{R}^{d}$ and for every weight $\omega$,

[TABLE]

Example 5.3**.**

We consider now the function $u(x)=e^{icx^{2}/2}$, for $x\in\mathbb{R}$ and $c\in\mathbb{R}\setminus\{0\}$. Observe that $u\in\mathcal{S}^{\prime}_{\omega}(\mathbb{R})$ for every $\omega$. Choosing as window function the Gaussian $\varphi(t)=e^{-t^{2}/2}$, that belongs to $\mathcal{S}_{\omega}(\mathbb{R})$ for every $\omega$, we have, as in Example 6.6 of [RW], that there exists $C>0$ such that

[TABLE]

Then, proceeding in a similar way as in the previous cases we have

[TABLE]

for every weight $\omega$.**

We observe that in the cases (5.87) and (5.88) the Gabor wave front set gives rougher information since it does not take into account translations and modulations, while for the case (5.89) it gives finer information, since it identifies the so-called instantaneous frequency, that is the only direction along which the time-frequency content of $u$ does not decay. For a comparison of the Gabor wave front set of the element considered in the previous examples with other type of global wave front set (at least in the frame of tempered distributions) we refer to [RW].

We observe now that in the previous examples the considered distributions have the same wave front set for every weight $\omega$. In general the Gabor wave front set may depend on $\omega$, as shown in the next example.

Example 5.4**.**

Let $\omega$ and $\sigma$ be two weight functions, such that $\omega(t)\leq\sigma(t)$ and $\mathcal{S}_{\sigma}(\mathbb{R}^{d})\cap\mathcal{D}(\mathbb{R}^{d})\subsetneq\mathcal{S}_{\omega}(\mathbb{R}^{d})\cap\mathcal{D}(\mathbb{R}^{d})$. We then fix a function $f\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$ with compact support such that $f\notin\mathcal{S}_{\sigma}(\mathbb{R}^{d})$. From Proposition 3.18 we have

[TABLE]

Fix now a window $\varphi_{0}\in\mathcal{S}_{\sigma}(\mathbb{R}^{d})$ with compact support such that $\varphi_{0}\equiv 1$ on $\mathop{\rm supp}\nolimits(f)$. From the definition of short-time Fourier transform, we then have that the orthogonal projection on $\mathbb{R}^{d}_{x}$ of the support of $V_{\varphi_{0}}f(x,\xi)$ is compact. Let now $z_{0}=(x_{0},\xi_{0})\in\mathbb{R}^{2d}$ with $x_{0}\neq 0$, and fix an open conic set containing $z_{0}$ of the form

[TABLE]

for $C>0$. We then have that $\overline{\Gamma\cap\mathop{\rm supp}\nolimits(V_{\varphi_{0}}f)}$ is compact, so the condition (3.25) is satisfied for every $\lambda>0$. Then $(x_{0},\xi_{0})\notin\mathop{\rm WF}\nolimits^{\prime}_{\sigma}(f)$ for every $x_{0}\neq 0$. Consider now a point of the type $(0,\xi_{0})$ with $\xi_{0}\neq 0$, $\xi_{0}\in\mathbb{R}^{d}$. From the fact that $\varphi_{0}\equiv 1$ on $\mathop{\rm supp}\nolimits(f)$, we have

[TABLE]

Since $f\notin\mathcal{S}_{\sigma}(\mathbb{R}^{d})$, we have that there exists $\lambda>0$ such that

[TABLE]

so (3.25) cannot be satisfied in an open conic set containing $(0,\xi_{0})$, and then $(0,\xi_{0})\in\mathop{\rm WF}\nolimits^{\prime}_{\omega}(f)$. We then have that

[TABLE]

in particular $\mathop{\rm WF}\nolimits^{\prime}_{\sigma}(f)\neq\mathop{\rm WF}\nolimits^{\prime}_{\omega}(f)$.**

Acknowledgments. The authors were partially supported by the INdAM-Gnampa Project 2016 “Nuove prospettive nell’analisi microlocale e tempo-frequenza”, by FAR 2013 (University of Ferrara) and by the project “Ricerca Locale - Analisi di Gabor, operatori pseudodifferenziali ed equazioni differenziali” (University of Torino). The research of the second author was partially supported by the project MTM2016-76647-P