An Introduction to Extended Gevrey Regularity

Abstract

Gevrey classes are the most common choice when considering the regularities of smooth functions that are not analytic. However, in various situations, it is important to consider smoothness properties that go beyond Gevrey regularity, for example, when initial value problems are ill-posed in Gevrey settings. In this paper, we consider a convenient framework for studying smooth functions that possess weaker regularity than any Gevrey function. Since the available literature on this topic is scattered, our aim is to provide an overview of extended Gevrey regularity, highlighting its most important features. Additionally, we consider related dual spaces of ultra distributions and review some results on micro-local analysis in the context of extended Gevrey regularity. We conclude the paper with a few selected applications that may motivate further study of the topic.

1. Introduction

Gevrey-type regularity was introduced in the study of fundamental solutions of the heat equation in [1] and subsequently used to describe regularities stronger than smoothness ($C^{\infty }$-regularity) and weaker than analyticity. This property turns out to be important in the general theory of linear partial differential equations, such as hypoellipticity, local solvability, and propagation of singularities, cf. [2]. In particular, the Cauchy problem for weakly hyperbolic linear partial differential equations (PDEs) can be well-posed in certain Gevrey classes, while at the same time being ill-posed in the class of analytic functions, as shown in [2,3].

Since there is a gap between Gevrey regularity and smoothness, it is important to study classes of smooth functions that do not belong to any Gevrey class. For example, Jézéquel [4] proved that the trace formula for Anosov flows in dynamical systems holds for certain intermediate regularity classes, and Cicognani and Lorenz used a different intermediate regularity when studying strictly hyperbolic equations in [5].

A systematic study of smoothness that goes beyond any Gevrey regularity was proposed in [6,7]. This was accomplished by introducing two-parameter dependent sequences of the form ${\left(p^{\tau p^{\sigma }}\right)}_p$, where $\tau >0$, $\sigma >1$. These sequences give rise to classes of ultradifferentiable functions ${\mathcal{E}}_{\tau ,\sigma }\left({\mathbb{R}}^d\right)$, which differ from classical Carleman classes $C^L\left({\mathbb{R}}^d\right)$ (cf. [8]), which are larger than Jézéquel’s classes, and which go beyond Komatsu’s approach to ultradifferentiable functions as described in, for example, [9]. These classes, called Pilipović–Teofanov–Tomić classes in [10], are a prominent example of the generalized matrix approach to ulradifferentiable functions. However, they provide asymptotic estimates in terms of the Lambert functions, which have proven to be useful in various contexts, as discussed in [5,11,12].

Different aspects of the so-called extended Gevrey regularity, i.e., the regularity of ultradifferentiable functions from ${\mathcal{E}}_{\tau ,\sigma }\left({\mathbb{R}}^d\right)$, have been studied in a dozen papers published in the last decade. Our aim is to offer a self-contained introduction to the subject and illuminate its main features. We provide proofs that, in general, simplify and complement those in the existing literature. Additionally, we present some new results, such as Proposition 2, Proposition 4, and Theorem 1 for the Beurling case, as well as Theorem 3.

This survey begins with preliminary Section 2, which covers the main properties of defining sequences, the Lambert function, and the associated function to a given sequence. We emphasize the remarkable connection between the associated function and the Lambert W function (see Theorem 1), which provides an elegant formulation of decay properties of the (short-time) Fourier transform of $f\in {\mathcal{E}}_{\tau ,\sigma }\left({\mathbb{R}}^d\right)$, as demonstrated in Proposition 5 and Corollary 2. In Section 3, we introduce the extended Gevrey classes ${\mathcal{E}}_{\tau ,\sigma }\left({\mathbb{R}}^d\right)$ and the corresponding spaces of ultradistributions. We then present their main properties, such as inverse closedness (Theorem 1) and the Paley–Wiener type theorem (Theorem 3).

In Section 4, we give an application of extended Gevrey regularity in micro-local analysis. More precisely, we introduce wave-front sets, which detect singularities that are “stronger” than classical $C^{\infty }$ singularities and, at the same time, “weaker” than any Gevrey-type singularity.

To provide a flavor of possible applications of extended Gevrey regularity, in Section 5, we briefly outline some results from [5,10]. More precisely, we present a result from [5] concerning the well-posedness of strictly hyperbolic equations in ${\mathcal{E}}_{1,2}\left({\mathbb{R}}^d\right)$, and observations from [10], where the extended Gevrey classes are referred to as Pilipović–Teofanov–Tomić classes and are considered within the extended matrix approach to ultradifferentiable classes.

We end this section by introducing some notation that will be used in the sequel.

Notation

We use the standard notation: $\mathbb{N}$, ${\mathbb{N}}_0$, $\mathbb{Z}$, $\mathbb{R}$, ${\mathbb{R}}_{+}$, $\mathbb{C}$, denote sets of positive integers, non-negative integers, integers, real numbers, positive real numbers and complex numbers, respectively. The length of a multi-index $\alpha =({\alpha }_1,\cdots ,{\alpha }_d)\in {\mathbb{N}}_0^d$ is denoted by $|\alpha |={\alpha }_1+{\alpha }_2+\cdots +{\alpha }_d$ and $\alpha !:={\alpha }_1!\cdots {\alpha }_d!$. For $x=(x_1,\cdots ,x_d)\in {\mathbb{R}}^d$ we denote: $|x|:={\left(x_1^2+\dots +x_d^2\right)}^{1/2},$$x^{\alpha }:={\prod }_{j=1}^d{x}_j^{{\alpha }_j},$ and $D^{\alpha }=D_x^{\alpha }:=D_1^{{\alpha }_1}\cdots D_d^{{\alpha }_d}$, where ${\displaystyle D_j^{{\alpha }_j}:={\left(-\frac{1}{2\pi i}\frac{\partial }{\partial x_j}\right)}^{{\alpha }_j},}$$j=1,\cdots ,d$. Throughout the paper, we use the convention $0^0=1$.

We write $L^p\left({\mathbb{R}}^d\right)$, $1\le p\le \infty$ for the Lebesgue spaces; $\mathcal{S}\left({\mathbb{R}}^d\right)$ denotes the Schwartz space of infinitely smooth ($C^{\infty }\left({\mathbb{R}}^d\right)$) functions which, together with their derivatives, decay at infinity faster than any inverse polynomial. By ${\mathcal{S}}^{\prime }\left({\mathbb{R}}^d\right)$ we denote the dual of $\mathcal{S}\left({\mathbb{R}}^d\right)$, the space of tempered distributions, and ${\mathcal{D}}^{\prime }\left({\mathbb{R}}^d\right)$ is the dual of $\mathcal{D}\left({\mathbb{R}}^d\right)=C_0^{\infty }\left({\mathbb{R}}^d\right)$, the space of compactly supported infinitely smooth functions.

We use brackets $\langle f,g\rangle$ to denote the extension of the inner product $\langle f,g\rangle =\int f\left(t\right)\overline{g\left(t\right)}dt$ on $L^2\left({\mathbb{R}}^d\right)$ to the dual pairing between a test function space $\mathcal{A}$ and its dual ${\mathcal{A}}^{\prime }$: ${\mathcal{A}}^{\prime }{\langle ·,\overline{·}\rangle }_{\mathcal{A}}=(·,·).$

The notation $f=O\left(g\right)$ means that $|f(x)|\le C|g(x)|$ for some $C>0$ and all x in the intersection of the domains for f and g. If $f=O\left(g\right)$ and $g=O\left(f\right)$, then we write $f\asymp g$.

The Fourier transform of $f\in L^1\left({\mathbb{R}}^d\right)$ given by

$$\widehat{f}\left(\xi \right):={\int }_{{\mathbb{R}}^d}f\left(x\right)e^{-2\pi ix·\xi }dx,\xi \in {\mathbb{R}}^d,$$

extends to $L^2\left({\mathbb{R}}^d\right)$ by the standard approximation procedure.

The convolution between $f,g\in L^1\left({\mathbb{R}}^d\right)$ is given by $(f\ast g)\left(t\right)=\int f\left(x\right)g(t-x)dx$.

Translation, modulation, and dilation operators, T, M, and D, respectively, when acting on $f\in L^2\left({\mathbb{R}}^d\right)$ are defined by

$$T_{x}f(·)=f(·-x)\mathrm{and}{M}_{x}f(·)=e^{2\pi ix·}f(·),D_{a}f(·)=\frac{1}{a}f\left(\frac{·}{a}\right),$$

$x\in {\mathbb{R}}^d$, $a>0$. Then, for $f,g\in L^2\left({\mathbb{R}}^d\right)$ the following relations hold:

$$M_y{T}_x=e^{2\pi ix·y}{T}_x{M}_y,\widehat{\left(T_{x}f\right)}=M_{-x}\widehat{f},\widehat{\left(M_{x}f\right)}=T_x\widehat{f},x,y\in {\mathbb{R}}^d.$$

The Fourier transform, convolution, T, M, and D are extended to other spaces of functions and distributions in a natural way.

2. Preliminaries

2.1. Defining Sequences via Komatsu

Komatsu’s approach to the theory of ultradistributions (see [9]) is based on sequences of positive numbers $\left(M_p\right)={\left(M_p\right)}_{p\in {\mathbb{N}}_0}$, $M_0=1$, which satisfy some of the following conditions:

• $(M.1)$ (logarithmic convexity)

  $$M_p^2\le M_{p-1}{M}_{p+1},p\in \mathbb{N};$$
• $(M.2)$ (stability under the action of ultradifferentiable operators/convolution)

  $$(\exists A,B>0)M_{p+q}\le A{B}^{p+q}{M}_p{M}_q,p,q\in {\mathbb{N}}_0;$$
• ${(M.2)}^{\prime }$ (stability under the action of differentiable operators)

  $$(\exists A,B>0)M_{p+1}\le A{B}^p{M}_p,p\in {\mathbb{N}}_0;$$
• $(M.3)$ (strong non-quasi-analyticity)

  $$(\exists A>0)\sum _{q=p+1}^{\infty }\frac{M_{q-1}}{M_q}\le Ap\frac{M_p}{M_{p+1}},p\in {\mathbb{N}}_0;$$
• ${(M.3)}^{\prime }$ (non-quasi-analyticity)

  $$\sum _{p=1}^{\infty }\frac{M_{p-1}}{M_p}<\infty .$$

Note, that $(M.2)\Rightarrow {(M.2)}^{\prime }$, and $(M.3)\Rightarrow {(M.3)}^{\prime }$. In addition, $(M.1)$ implies $M_p{M}_q\le M_{p+q}$, $p,q\in {\mathbb{N}}_0$.

Let $\left(M_p\right)$ be a positive monotone increasing sequence that satisfies $(M.1)$. Then ${(M_p/p!)}^{1/p},$ is an almost increasing sequence if there exists $C>0$ such that

$${\left(\frac{M_p}{p!}\right)}^{1/p}\le C{\left(\frac{M_q}{q!}\right)}^{1/q},p\le q.$$

(1)

This property is related to inverse closedness in $C^{\infty }\left({\mathbb{R}}^d\right)$, see [13]. Note, that if ${(M_p/p!)}^{1/p}$ is an almost increasing sequence, then ${lim}_{p\to \infty }{M}_p^{1/p}=\infty$. Indeed, for $p=1$ and all $q\ge 1$ we obtain ${(M_q/q!)}^{1/q}\ge M_1{C}^{-1}$, i.e., $M_q^{1/q}\ge M_1{C}^{-1}q{!}^{1/q}$. Now $\left(M_p^{1/p}\right)$ diverges to infinity by Stirling’s formula.

The Gevrey sequence $M_p=p{!}^s$, $p\in {\mathbb{N}}_0,$$s>1$ satisfies $(M.1)$, $(M.2)$, and $(M.3)$. It is also an almost increasing sequence.

If $\left(M_p\right)$ and $\left(N_p\right)$ satisfy $(M.1)$, then we write $M_p\subset N_p$ if there are the constants $A>0$ and $B>0$ (independent on p) such that

$$M_p\le A{B}^p{N}_p,p\in {\mathbb{N}}_0.$$

(2)

If, instead, for each $B>0$ there exists $A>0$ such that (2) holds, then we write

$$M_p\prec N_p.$$

Assume that $\left(M_p\right)$ satisfies $(M.1)$ and ${(M.3)}^{\prime }$. Then, $p!\prec M_p$.

Let $\mathcal{R}$ denote the set of all sequences of positive numbers monotonically increasing to infinity. For a given sequence $\left(M_p\right)$ and $\left(r_p\right)\in \mathcal{R}$ we consider

$$N_0=1,N_p=M_p{r}_1{r}_2\cdots r_p=M_p\prod _{j=1}^p{r}_j,p\in \mathbb{N}.$$

It is easy to see that if $\left(M_p\right)$ satisfies $(M.1)$ and ${(M.3)}^{\prime }$, then $\left(N_p\right)$ satisfies $(M.1)$ and ${(M.3)}^{\prime }$ as well. In addition, one can find $\left({\tilde{r}}_p\right)\in \mathcal{R}$ so that $\left(M_p{\prod }_{j=1}^p{\tilde{r}}_j\right)$ satisfies $(M.2)$ if $\left(M_p\right)$ does. This follows from the next lemma.

Lemma 1. 

Let $\left(r_p\right)\in \mathcal{R}$ be given. Then, there exists $\left({\tilde{r}}_p\right)\in \mathcal{R}$ such that ${\tilde{r}}_p\le r_p$, $p\in \mathbb{N}$, and

$$\prod _{j=1}^{p+q}{\tilde{r}}_j\le 2^{p+q}\prod _{j=1}^p{\tilde{r}}_j\prod _{j=1}^q{\tilde{r}}_j,p,q\in \mathbb{N}.$$

(3)

Proof. 

It is enough to consider the sequence $\left({\tilde{r}}_p\right)$ given by ${\tilde{r}}_1=r_1$ and inductively

$${\tilde{r}}_{j+1}=min\left\{r_{j+1},\frac{j+1}{j}{\tilde{r}}_j\right\},j\in \mathbb{N}.$$

Then, $\left({\tilde{r}}_p\right)\in \mathcal{R}$ and (3) holds. We refer to [14] (Lemma 2.3) for details. □

2.2. Defining Sequences for Extended Gevrey Regularity

To extend the class of Gevrey-type ultradifferentiable functions we consider two-parameter sequences of the form $M_p^{\tau ,\sigma }=p^{\tau p^{\sigma }}$, $p\in \mathbb{N},$$\tau >0$, $\sigma >1$.

From Stirling’s formula, for any (fixed) $s,\sigma >1$, and $\tau >0$, we can find $C>0$ (independent of p) such that

$$sp\le C\tau p^{\sigma },p\in {\mathbb{N}}_0,$$

and so it follows that there exists $C_1>0$ such that

$$p{!}^s\le C_1{p}^{\tau p^{\sigma }}.$$

(4)

The main properties of $\left(M_p^{\tau ,\sigma }\right)$ are collected in the next lemma (cf. [6] (Lemmas 2.2 and 3.1)). The proof is given in the Appendix A.

Lemma 2. 

Let $\tau >0$, $\sigma >1$, $M_0^{\tau ,\sigma }=1$, and $M_p^{\tau ,\sigma }=p^{\tau p^{\sigma }}$, $p\in \mathbb{N}$. Then, the following properties hold:

• $(M.1)$  ${\left(M_p^{\tau ,\sigma }\right)}^2\le M_{p-1}^{\tau ,\sigma }{M}_{p+1}^{\tau ,\sigma },p\in \mathbb{N},$
• $\widetilde{(M.2)}$ $M_{p+q}^{\tau ,\sigma }\le C^{p^{\sigma }+q^{\sigma }}{M}_p^{\tau 2^{\sigma -1},\sigma }{M}_q^{\tau 2^{\sigma -1},\sigma },p,q\in {\mathbb{N}}_0,$ for some constant $C\ge 1$,
• $\widetilde{{(M.2)}^{\prime }}$ $M_{p+1}^{\tau ,\sigma }\le C^{p^{\sigma }}{M}_p^{\tau ,\sigma },p\in {\mathbb{N}}_0,$ for some constant $C\ge 1$,
• ${(M.3)}^{\prime }$  ${\displaystyle \sum _{p=1}^{\infty }\frac{M_{p-1}^{\tau ,\sigma }}{M_p^{\tau ,\sigma }}<\infty .}$

Remark 1. 

From the proof of $\widetilde{{(M.2)}^{\prime }}$ it follows that $\left(M_p^{\tau ,\sigma }\right)$ does not satisfy ${(M.2)}^{\prime }$, and consequently $(M.2)$ is violated as well. One might expect that instead, the sequence $\left(M_p^{\tau ,\sigma }\right)$ satisfies

$$M_{p+q}^{\tau ,\sigma }\le C^{p^{\sigma }+q^{\sigma }}{M}_p^{\tau ,\sigma }{M}_q^{\tau ,\sigma },p,q\in {\mathbb{N}}_0,$$

(5)

for some constant $C>0$. However, if we assume that (5) holds for, e.g., $\tau =1$, then, for $p=q\ne 0$, we obtain

$${\left(2p\right)}^{{\left(2p\right)}^{\sigma }}\le {\left(Cp\right)}^{2p^{\sigma }},p\in \mathbb{N},$$

which gives

$$p^{(2^{\sigma }-2)}\le C^2{2}^{-2^{\sigma }},\mathit{for}\mathit{all}p\in \mathbb{N},$$

a contradiction. Thus, $\widetilde{(M.2)}$ is a suitable alternative to $(M.2)$ when considering $\left(M_p^{\tau ,\sigma }\right)$.

Let $M_p^{\tau ,\sigma }=p^{\tau p^{\sigma }}$, $p\in \mathbb{N},$$\tau >0$, $\sigma >1$, and $\left(r_p\right)\in \mathcal{R}$. If $\left({\tilde{r}}_p\right)\in \mathcal{R}$ is chosen as in Lemma 1, then the sequence $\left(N_p\right)$ given by

$$N_0=1,N_p=M_p^{\tau ,\sigma }\prod _{j=1}^p{\tilde{r}}_j,p\in \mathbb{N},$$

satisfies $(M.1)$, $\widetilde{(M.2)}$, $\widetilde{{(M.2)}^{\prime }}$, and ${(M.3)}^{\prime }$.

We note that if $M_p^{\tau ,\sigma }=p^{\tau p^{\sigma }}$, $p\in \mathbb{N},$$\tau >0$, $\sigma >1$, then the sequence ${\left(\frac{M_p^{\tau ,\sigma }}{p^p}\right)}^{1/p},$$p\in \mathbb{N},$ is an almost increasing sequence since ${\left(\frac{M_p^{\tau ,\sigma }}{p^p}\right)}^{1/p}=p^{\tau p^{\sigma -1}-1}$ and $p^{\tau p^{\sigma -1}-1}<q^{\tau q^{\sigma -1}-1},$ for all $p,q\in \mathbb{N},$ such that $\lceil {(1/\tau )}^{1/(\sigma -1)}\rceil <p<q.$ Here, $\lceil x\rceil$ denotes the smallest integer greater than or equal to x.

2.3. The Lambert Function

The Lambert W function is defined as the inverse of $z{e}^z$, $z\in \mathbb{C}$. By $W\left(x\right)$, we denote the restriction of its principal branch to $[0,\infty )$. It is used as a convenient tool to describe asymptotic behavior in different contexts. We refer to [15] for a review of some applications of the Lambert W function in pure and applied mathematics, and to the recent monograph [16] for more details and generalizations. It is noteworthy that the Lambert function describes the precise asymptotic behavior of the associated function to the sequence $\left(M_p^{\tau ,\sigma }\right)$. This fact was first observed in [17].

Some basic properties of the Lambert function W are given below:

• $\left(W1\right)$ $W\left(0\right)=0$, $W\left(e\right)=1$, $W\left(x\right)$ is continuous, increasing and concave on $[0,\infty )$,
• $\left(W2\right)$ $W\left(x{e}^x\right)=x$ and $x=W\left(x\right)e^{W\left(x\right)}$, $x\ge 0$,
• $\left(W3\right)$ The following estimates hold:

  $$lnx-ln(lnx)\le W\left(x\right)\le lnx-\frac{1}{2}ln(lnx),x\ge e,$$

  (6)

  with the equality if and only if $x=e$, see [18].

Note, that $\left(W2\right)$ implies

$$W(xlnx)=lnx,x>1.$$

By using $\left(W3\right)$ we obtain

$$W\left(x\right)\sim lnx,x\to \infty ,$$

and, therefore,

$$W\left(Cx\right)\sim W\left(x\right),x\to \infty ,$$

for any $C>0$. We refer to [15,16] for more details about the Lambert W function.

2.4. Associated Functions

Let $\left(M_p\right)$ be an increasing sequence of positive numbers which satisfies $(M.1)$, and $M_0=1$. Then, the Carleman associated function to the sequence $\left(M_p\right)$ is defined by

$$\mu \left(h\right)=\underset{p\in {\mathbb{N}}_0}{inf}{h}^{-p}{M}_p,h>0.$$

(7)

This function is introduced in the study of quasi-analytic functions, see, e.g., [19]. We use the notation from [20].

In Komatsu’s treatise of ultradistributions [9], the associated function to $\left(M_p\right)$ is instead given by

$$T\left(h\right)=\underset{p\in {\mathbb{N}}_0}{sup}ln\frac{h^p}{M_p},h>0.$$

(8)

Lemma 3. 

Let $\left(M_p\right)$ be an increasing sequence of positive numbers that satisfies $(M.1)$, and $M_0=1$, and let the functions μ and T be given by (7) and (8), respectively. Then,

$$\mu \left(h\right)=e^{-T\left(h\right)},h>0.$$

(9)

Proof. 

Clearly,

$$\begin{array}{ccc}\hfill T\left(h\right)=\underset{p\in {\mathbb{N}}_0}{sup}ln\left(h^p{M}_p^{-1}\right)& =& \underset{p\in {\mathbb{N}}_0}{sup}(-ln\left(h^{-p}{M}_p\right))\hfill \\ & =& -\underset{p\in {\mathbb{N}}_0}{inf}(ln\left(h^{-p}{M}_p\right))=-ln\left(\underset{p\in {\mathbb{N}}_0}{inf}\left(h^{-p}{M}_p\right)\right)\hfill \\ & =& -ln\mu \left(h\right),h>0,\hfill \end{array}$$

which is (9). □

When $M_p=p^{tp}$, $p\in \mathbb{N},$$t>1$, $M_0=1$, an explicit calculation gives

$$T\left(h\right)=\frac{s}{e}{h}^{\frac{1}{s}},h>0.$$

Thus, (9) implies that there exist constants $k>0$, and $C>0$ such that

$$e^{-k{h}^{\frac{1}{s}}}\le \mu \left(h\right)\le C{e}^{-k{h}^{\frac{1}{s}}},h>0,$$

see also [20] (Chapter IV, Section 2.1). We note that $\left(p^{tp}\right)$ can be used in (13) instead of the Gevrey sequence $\left(p{!}^t\right)$, $t>1$, to define Gevrey spaces ${\mathcal{G}}_t\left({\mathbb{R}}^d\right)$, $t>1$, see [2] (Proposition 1.4.2).

By using (8) we define the associated function to the sequence $M_p^{\tau ,\sigma }=p^{\tau p^{\sigma }}$, $p\in \mathbb{N},$$\tau >0$, $\sigma >1$, as follows:

$$T_{\tau ,\sigma }\left(h\right)=\underset{p\in {\mathbb{N}}_0}{sup}ln\frac{h^p}{M_p^{\tau ,\sigma }},h>0.$$

(10)

It is a remarkable fact that $T_{\tau ,\sigma }\left(h\right)$ can be expressed via the Lambert W function.

Theorem 1. 

Let $\tau >0$, $\sigma >1$, $M_0^{\tau ,\sigma }=1$, and $M_p^{\tau ,\sigma }=p^{\tau p^{\sigma }}$, $p\in \mathbb{N}$, and let $T_{\tau ,\sigma }\left(h\right)$ be given by (10). Then

$$T_{\tau ,\sigma }\left(h\right)\asymp {\tau }^{-\frac{1}{\sigma -1}}\frac{{ln}^{\frac{\sigma }{\sigma -1}}\left(h\right)}{W^{\frac{1}{\sigma -1}}(ln\left(h\right))},h>1,$$

(11)

where the hidden constants depend on σ only.

Proof. 

The proof follows from estimates (30) in the proof of [21] (Proposition 2). More precisely, it can be shown that

$$B_{\sigma }{\tau }^{-\frac{1}{\sigma -1}}\frac{{ln}^{\frac{\sigma }{\sigma -1}}\left(h\right)}{W^{\frac{1}{\sigma -1}}(ln\left(h\right))}+{\tilde{B}}_{\tau ,\sigma }\le T_{\tau ,\sigma }\left(h\right)\le A_{\sigma }{\tau }^{-\frac{1}{\sigma -1}}\frac{{ln}^{\frac{\sigma }{\sigma -1}}\left(h\right)}{W^{\frac{1}{\sigma -1}}(ln\left(h\right))}+{\tilde{A}}_{\tau ,\sigma },$$

(12)

for $h>1$, and suitable constants $A_{\sigma },B_{\sigma },{\tilde{A}}_{\tau ,\sigma },{\tilde{B}}_{\tau ,\sigma }>0$. □

We also noticed that $\left(W3\right)$ (from Section 2.3) implies

$$T_{\tau ,\sigma }\left(h\right)\asymp {\left(\frac{{ln}^{\sigma }\left(h\right)}{\tau ln(ln(h\left)\right)}\right)}^{\frac{1}{\sigma -1}},\mathrm{for}h>1.$$

2.5. Associated Function as a Weight Function

The approach to ultradifferentiable functions via defining sequences is equivalent to the Braun–Meise–Taylor approach based on weight functions, when the defining sequences satisfy conditions $(M.1),$$(M.2)$ and $(M.3),$ see [22,23]. Since $M_p^{\tau ,\sigma }=p^{\tau p^{\sigma }},$$p\in \mathbb{N},$ does not satisfy $(M.2),$ to compare the two approaches in [21], the authors used the technique of weighted matrices, see [24]. One of the main results from [21] can be stated as follows.

Proposition 1. 

Let $\tau >0$, $\sigma >1$, $M_0^{\tau ,\sigma }=1$, and $M_p^{\tau ,\sigma }=p^{\tau p^{\sigma }},$$p\in \mathbb{N}$, and let $T_{\tau ,\sigma }\left(h\right)$ be the associated function to the sequence $\left(M_p^{\tau ,\sigma }\right)$. Then, $T_{\tau ,\sigma }\left(h\right)\asymp \omega \left(h\right)$, where ω is a weight function.

Recall that a weight function $\omega$ is a non-negative, continuous, even and increasing function defined on ${\mathbb{R}}_{+}\cup \left\{0\right\}$, $\omega \left(0\right)=0$, if the following conditions hold:

• ($\alpha$) ${\displaystyle \omega \left(2t\right)=O\left(\omega \right(t\left)\right),t\to \infty ,}$
• ($\beta$) ${\displaystyle \omega \left(t\right)=O\left(t\right),t\to \infty ,}$
• ($\gamma$) ${\displaystyle lnt=o\left(\omega \right(t\left)\right),t\to \infty ,}$ i.e., ${\displaystyle \underset{t\to \infty }{lim}\frac{lnt}{\omega \left(t\right)}=0}$,
• ($\delta$) ${\displaystyle \phi \left(t\right)=w\left(e^t\right)}$ is convex.

Some classical examples of weight functions are as follows:

$$\omega \left(t\right)={ln}^s(1+|t|),\omega \left(t\right)=\frac{|t|}{{ln}^{s-1}(e+|t|)},s>1,t\in \mathbb{R}.$$

Moreover, $\omega \left(t\right)={|t|}^s$ is a weight function if and only if $0<s\le 1$.

We refer to [24] for the weighted matrices approach to ultradifferentiable functions. It is introduced in order to treat both Braun–Meise–Taylor and Komatsu methods in a unified way, see also Section 5.2.

3. Extended Gevrey Regularity

3.1. Extended Gevrey Classes and Their Dual Spaces

Recall that the Gevrey space ${\mathcal{G}}_t\left({\mathbb{R}}^d\right)$, $t>1$, consists of functions $\varphi \in C^{\infty }\left({\mathbb{R}}^d\right)$ such that for every compact set $K\subset \subset {\mathbb{R}}^d$ there are constants $h>0$ and $C_K>0$ satisfying

$$|{\partial }^{\alpha }\varphi \left(x\right)|\le C_K{h}^{|\alpha |}|\alpha |{!}^t,$$

(13)

for all $x\in K$ and for all $\alpha \in {\mathbb{N}}_0^d$.

In a similar fashion we introduce new classes of smooth functions by using defining sequences $M_p^{\tau ,\sigma }=p^{\tau p^{\sigma }}$, $p\in \mathbb{N}$, $\tau >0$, $\sigma >1$.

Definition 1. 

Let there be given $\tau >0$, $\sigma >1$, and let $M_p^{\tau ,\sigma }=p^{\tau p^{\sigma }}$, $p\in \mathbb{N}$, $M_0^{\tau ,\sigma }=1$.

The extended Gevrey class of Roumieu type ${\mathcal{E}}_{\{\tau ,\sigma \}}\left({\mathbb{R}}^d\right)$ is the set of all $\varphi \in C^{\infty }\left({\mathbb{R}}^d\right)$ such that for every compact set $K\subset \subset {\mathbb{R}}^d$ there are constants $h>0$ and $C_K>0$ satisfying

$$|{\partial }^{\alpha }\varphi \left(x\right)|\le C_K{h}^{{|\alpha |}^{\sigma }}{M}_{|\alpha |}^{\tau ,\sigma },$$

(14)

for all $x\in K$ and for all $\alpha \in {\mathbb{N}}_0^d.$

The extended Gevrey class of Beurling type ${\mathcal{E}}_{(\tau ,\sigma )}\left({\mathbb{R}}^d\right)$ is the set of all $\varphi \in C^{\infty }\left({\mathbb{R}}^d\right)$ such that for every compact set $K\subset \subset {\mathbb{R}}^d$ and for all $h>0$ there is a constant $C_{K,h}>0$ satisfying

$$|{\partial }^{\alpha }\varphi \left(x\right)|\le C_{K,h}{h}^{{|\alpha |}^{\sigma }}{M}_{|\alpha |}^{\tau ,\sigma },$$

(15)

for all $x\in K$ and for all $\alpha \in {\mathbb{N}}_0^d.$

The Roumieu space ${\mathcal{E}}_{\{\tau ,\sigma \}}\left({\mathbb{R}}^d\right)$ is a countable projective limit of countable inductive limits of Banach spaces (i.e., a so-called (PLB)-space), and the Beurling space ${\mathcal{E}}_{(\tau ,\sigma )}\left({\mathbb{R}}^d\right)$ is a Fréchet space, we refer to [6] for details. In particular, they are nuclear spaces, see [6] (Theorem 3.1).

Note, that (4), (13)–(15) imply

$${\cup }_{t>1}{\mathcal{G}}_t\left({\mathbb{R}}^d\right)\hookrightarrow {\mathcal{E}}_{(\tau ,\sigma )}\left({\mathbb{R}}^d\right)\hookrightarrow {\mathcal{E}}_{\{\tau ,\sigma \}}\left({\mathbb{R}}^d\right),$$

where ↪ denotes continuous and dense inclusion.

The subspace of functions $\varphi \in {\mathcal{E}}_{\{\tau ,\sigma \}}\left({\mathbb{R}}^d\right)$ ($\varphi \in {\mathcal{E}}_{(\tau ,\sigma )}\left({\mathbb{R}}^d\right)$) whose support is contained in some compact set is denoted by ${\mathcal{D}}_{\{\tau ,\sigma \}}\left({\mathbb{R}}^d\right)$ (${\mathcal{D}}_{(\tau ,\sigma )}\left({\mathbb{R}}^d\right)$).

We use the abbreviated notation $\tau ,\sigma$ for $\{\tau ,\sigma \}$ or $(\tau ,\sigma )$ to denote ${\mathcal{E}}_{\tau ,\sigma }\left({\mathbb{R}}^d\right)={\mathcal{E}}_{\{\tau ,\sigma \}}\left({\mathbb{R}}^d\right)$ or ${\mathcal{E}}_{\tau ,\sigma }\left({\mathbb{R}}^d\right)={\mathcal{E}}_{(\tau ,\sigma )}\left({\mathbb{R}}^d\right)$, and similarly for ${\mathcal{D}}_{\tau ,\sigma }\left({\mathbb{R}}^d\right)$.

Next, we give an equivalent description of extended Gevrey classes by using sequences from $\mathcal{R}$, see Section 2.2. We note that such descriptions are important when dealing with integral transforms of ultradifferentiable functions and related ultradistributions, cf. [14,25,26]. The result follows from a lemma which is a modification of [26] (Lemma 3.4), and [25] (Lemma 2.2.1).

Put $\lfloor x\rfloor :=max\{m\in {\mathbb{N}}_0:m\le x\}$ (the greatest integer part of $x\in {\mathbb{R}}_{+}$).

Lemma 4. 

Let there be given $\sigma >1$, a sequence of positive numbers $\left(a_p\right)$, $\left(r_j\right)\in \mathcal{R}$, and put

$$R_{0,\sigma }=1,R_{p,\sigma }:=\prod _{j=1}^{\lfloor p^{\sigma }\rfloor }{r}_j,p\in \mathbb{N}.$$

(16)

Then, the following is true.

(i) 

There exists $h>0$ such that

$$sup\left\{\frac{a_p}{h^{p^{\sigma }}}:p\in {\mathbb{N}}_0\right\}<\infty ,$$

if and only if

$$sup\left\{\frac{a_p}{R_{p,\sigma }}:p\in {\mathbb{N}}_0\right\}<\infty ,\mathit{for}\mathit{any}\left(r_j\right)\in \mathcal{R}.$$

(17)

(ii) 

There exists $\left(r_j\right)\in \mathcal{R}$ such that

$$sup\left\{R_{p,\sigma }{a}_p:p\in {\mathbb{N}}_0\right\}<\infty ,$$

if and only if

$$sup\left\{h^{p^{\sigma }}{a}_p:p\in {\mathbb{N}}_0\right\}<\infty ,\mathit{for}\mathit{every}h>0.$$

(18)

The proof of Lemma 4 is given in the Appendix A.

Note, that in (14) and (15) we could put $h^{\lfloor |\alpha {|}^{\sigma }\rfloor }$ instead of $h^{{|\alpha |}^{\sigma }}$ (this follows from the simple inequality $\lfloor p^{\sigma }\rfloor \le p^{\sigma }\le 2\lfloor p^{\sigma }\rfloor$, $p\in \mathbb{N}$).

Proposition 2. 

Let there be given $\tau >0$, $\sigma >1$, and let $M_p^{\tau ,\sigma }=p^{\tau p^{\sigma }}$, $p\in \mathbb{N}$, $M_0^{\tau ,\sigma }=1$. Then, the following is true:

(i) 

$\varphi \in {\mathcal{E}}_{\{\tau ,\sigma \}}\left({\mathbb{R}}^d\right)$ if and only if for every compact set $K\subset \subset {\mathbb{R}}^d$, and for any $\left(r_j\right)\in \mathcal{R}$ and $R_{p,\sigma }$ given by (16), there exists $C_{K,\left(r_j\right)}>0$ such that

$$|{\partial }^{\alpha }\varphi \left(x\right)|\le C_{K,\left(r_j\right)}{R}_{\lfloor |\alpha {|}^{\sigma }\rfloor ,\sigma }{M}_{|\alpha |}^{\tau ,\sigma },$$

for all $x\in K$, and all $\alpha \in {\mathbb{N}}_0^d$.

(ii) 

$\varphi \in {\mathcal{E}}_{(\tau ,\sigma )}\left({\mathbb{R}}^d\right)$ if and only if for every compact set $K\subset \subset {\mathbb{R}}^d$ there is a sequence $\left(r_j\right)\in \mathcal{R}$ and a constant $C_K>0$ satisfying

$$|{\partial }^{\alpha }\varphi \left(x\right)|\le C_K\frac{M_{|\alpha |}^{\tau ,\sigma }}{R_{\lfloor |\alpha {|}^{\sigma }\rfloor ,\sigma }}$$

for all $x\in K$ and for all $\alpha \in {\mathbb{N}}_0^d,$ where $R_{p,\sigma }$ given by (16).

Proposition 2 follows from Lemma 4.

We end this subsection by introducing spaces of ulradistributions as dual spaces of ${\mathcal{E}}_{\tau ,\sigma }\left({\mathbb{R}}^d\right)$, $\tau >0$, $\sigma >1$. In Section 3.5 we will prove a Paley–Wiener type theorem for such ultradistributions.

Definition 2. 

Let $\tau >0$ and $\sigma >1$. Then, $u\in {\mathcal{E}}_{\{\tau ,\sigma \}}^{\prime }\left({\mathbb{R}}^d\right)$ (resp. $u\in {\mathcal{E}}_{(\tau ,\sigma )}^{\prime }\left({\mathbb{R}}^d\right)$) if there exists a compact set K in ${\mathbb{R}}^d$ and for every $\epsilon >0$ there exists a constant $C>0$ (resp. there exist constants $\epsilon ,C>0$) such that

$$|(u,\phi )|\le C\underset{\alpha \in {\mathbb{N}}_0^d,x\in K}{sup}\frac{|D^{\alpha }\phi \left(x\right)|}{{\epsilon }^{{|\alpha |}^{\sigma }}{|\alpha |}^{{\tau |\alpha |}^{\sigma }}},\forall \phi \in {\mathcal{E}}_{\tau ,\sigma }\left({\mathbb{R}}^d\right),$$

(19)

and $(·,·)$ denotes the standard dual pairing.

In a similar way ${\mathcal{D}}_{\tau ,\sigma }^{\prime }\left({\mathbb{R}}^d\right)$ is the dual space of ${\mathcal{D}}_{\tau ,\sigma }\left({\mathbb{R}}^d\right)$.

3.2. Example of a Compactly Supported Function

The non-quasianalyticity condition ${(M.3)}^{\prime }$ provides the existence of nontrivial compactly supported functions in ${\mathcal{E}}_{\tau ,\sigma }\left({\mathbb{R}}^d\right)$ which can be formulated as follows:

Proposition 3. 

Let $\sigma >1$. For every $a>0$ there exists ${\varphi }_a\in {\bigcap }_{\tau >0}{\mathcal{D}}_{\tau ,\sigma }\left({\mathbb{R}}^d\right)$ such that ${\varphi }_a\ge 0$, $supp{\varphi }_a\subset {[-a,a]}^d$, and ${\int }_{{\mathbb{R}}^d}{\varphi }_a\left(x\right)dx=1$.

Proof. 

We give a proof when $d=1$, and for $d\ge 2$ the proof follows by taking the tensor product.

Since ${\mathcal{D}}_{\tau ,\sigma }\left(\mathbb{R}\right)$, $\tau >0$ is closed under dilation and multiplication by a constant, it is enough to show the result for $a=1$, and set ${\varphi }_1=\varphi$.

From

$$\sum _{p=1}^{\infty }\frac{1}{{\left(2(p+1)\right)}^{\frac{1}{m}{p}^{\sigma -1}}}<\infty$$

(20)

for any $m\in \mathbb{N}$ and any given $\sigma >1$, it follows that there exists a sequence of nonnegative integers $\left(N_m\right)$ such that

$$\sum _{p=N_m}^{\infty }\frac{1}{{\left(2(p+1)\right)}^{\frac{1}{m}{p}^{\sigma -1}}}<\frac{1}{2^m}.$$

Thus, the sequence $a_p$, $p\in {\mathbb{N}}_0$, given by

$$a_p:=\frac{1}{{\left(2(p+1)\right)}^{\frac{1}{m}{p}^{\sigma -1}}},N_m\le p<N_{m+1},$$

satisfies

$$\sum _{p=N_1}^{\infty }{a}_p\le 1.$$

Let $f\in C^{\infty }\left(\mathbb{R}\right)$ be a non-negative and even function such that $suppf\subset [-1,1]$, ${\int }_{-1}^{1}f\left(x\right)dx=1$, and set $f_a\left(x\right):=\frac{1}{a}f\left(\frac{x}{a}\right)$, $a>0$. Then, we define the sequence of functions $\left({\varphi }_p\right)$ by

$${\varphi }_p:=f_{a_{N_1}}\ast f_{a_{N_1+1}}\ast \cdots \ast f_{a_p},p\in {\mathbb{N}}_0.$$

Note, that

$$supp{\varphi }_p\subset [-1,1],{\int }_{-1}^1{\varphi }_p\left(x\right)dx=1,p\in {\mathbb{N}}_0,$$

$${\varphi }_p^{\left(n\right)}=f_{a_{N_1}}\ast \cdots \ast f_{a_{N_m}}\ast f_{a_{N_m+1}}^{\prime }\ast \cdots \ast f_{a_{N_m+n}}^{\prime }\ast f_{a_{N_m+n+1}}\ast \cdots \ast f_{a_p},$$

and there exists a constant $c>0$ such that

$$\parallel f_{a_p}^{\prime }{\parallel }_1=\frac{1}{a_p}{\int }_{\mathbb{R}}\frac{1}{a_p}\left|f^{\prime }\left(\frac{x}{a_p}\right)\right|dx\le \frac{c}{a_p}\le c{\left(2(p+1)\right)}^{\frac{1}{m}{p}^{\sigma -1}},$$

(21)

when $p\ge N_m$.

Let there be given $n\in {\mathbb{N}}_0$ and $\tau >0$. Then, we choose $m,p\in \mathbb{N}$ so that $1/m<\tau$, and $N_m+n<p$.

By using (21) and the fact that $\widetilde{{(M.2)}^{\prime }}$ implies

$$M_{p+q}^{\frac{1}{m},\sigma }\le {\tilde{C}}^{p^{\sigma }}{M}_p^{\frac{1}{m},\sigma }$$

for some $\tilde{C}=\tilde{C}\left(q\right)>0$, we obtain

$$\begin{array}{cc}\hfill |{\varphi }_p^{\left(n\right)}\left(x\right)|& \le c^n{2}^{\frac{1}{m}{\sum }_{k=1}^n{(N_m+k)}^{\sigma -1}}\prod _{k=1}^n{(N_m+k+1)}^{\frac{1}{m}{(N_m+k)}^{\sigma -1}}\hfill \\ \hfill & \le c^n{2}^{\frac{1}{m}{(N_m+n)}^{\sigma }}{(N_m+n+1)}^{\frac{1}{m}{(N_m+n+1)}^{\sigma }}\hfill \\ \hfill & \le C^{n^{\sigma }}{n}^{\tau n^{\sigma }},\hfill \end{array}$$

where C depends on $\tau$.

The sequence $\{{\varphi }_p^{\left(n\right)}\mid p=N_1,N_2,\cdots \}$ is a Cauchy sequence for every $n\in {\mathbb{N}}_0$. Thus, it converges to a function $\varphi$ that satisfies

$$|{\varphi }^{\left(n\right)}\left(x\right)|\le C^{n^{\sigma }}{n}^{\tau n^{\sigma }},$$

for every $\tau >0$, and all $n\in {\mathbb{N}}_0$. Therefore, $\varphi \in {\mathcal{D}}_{\tau ,\sigma }\left(\mathbb{R}\right)$ for every $\tau >0$, and by the construction $\varphi \ge 0$, $supp\varphi \subset [-1,1]$ and ${\int }_{\mathbb{R}}\varphi \left(x\right)dx=1$, which completes the proof. □

Remark 2. 

The construction in Proposition 3 provides a function in ${\bigcap }_{\tau >0}{\mathcal{D}}_{\tau ,\sigma }\left({\mathbb{R}}^d\right)$ which does not belong to any Gevrey class. This can be justified as follows. First, we note that the construction of ϕ in Proposition 3 can be modified to obtain ${\varphi }^{\left(n\right)}\left(0\right)={\delta }_{n,0}$ (Kronecker Delta), $n\in {\mathbb{N}}_0$ (cf. [7] (Example 3.1)). Then, we may take a so-called characteristic function ψ belonging to a Roumieu class of ultradifferentiable functions which is strictly larger than ${\cup }_{t>1}{\mathcal{G}}_t\left({\mathbb{R}}^d\right)$ and strictly smaller than ${\bigcap }_{\tau >0}{\mathcal{D}}_{\tau ,\sigma }\left({\mathbb{R}}^d\right)$ (see, e.g., [27] (Theorem 1)). In view of the product rule and Proposition 4 given below, it follows that $\psi \varphi \in {\bigcap }_{\tau >0}{\mathcal{D}}_{\tau ,\sigma }\left({\mathbb{R}}^d\right)\setminus {\cup }_{t>1}{\mathcal{G}}_t\left({\mathbb{R}}^d\right)$.

3.3. Algebra Property

We remark that the stability properties given in this and the next subsection are recently commented on in [10], by using completely different techniques. Our aim is to give here independent and direct proofs for the convenience of the reader.

Since $\left(M_p^{\tau ,\sigma }\right)$ satisfies properties $(M.1)$ and $\widetilde{{(M.2)}^{\prime }}$ we have the following.

Proposition 4. 

${\mathcal{E}}_{\tau ,\sigma }\left({\mathbb{R}}^d\right)$ is closed under the pointwise multiplication of functions and under the (finite order) differentiation.

Proof. 

Let us prove that ${\mathcal{E}}_{(\tau ,\sigma )}\left({\mathbb{R}}^d\right)$ is closed under pointwise multiplication, since its closedness under the differentiation follows from $\widetilde{{(M.2)}^{\prime }}$. We refer to [6] for the Roumieu case ${\mathcal{E}}_{\{\tau ,\sigma \}}\left({\mathbb{R}}^d\right)$.

Let $\varphi ,\psi \in {\mathcal{E}}_{(\tau ,\sigma )}\left({\mathbb{R}}^d\right)$. Let K be a compact subset of ${\mathbb{R}}^d$. Then, for every $h,k>0$ there exist constants $C_{K,h}>0$ and $C_{K,k}>0$ such that

$$\underset{x\in K}{sup}|{\partial }^{\alpha }\varphi \left(x\right)|\le C_{K,h}{h}^{{|\alpha |}^{\sigma }}{|\alpha |}^{{\tau |\alpha |}^{\sigma }},\mathrm{and}\underset{x\in K}{sup}|{\partial }^{\alpha }\psi \left(x\right)|\le C_{K,k}{k}^{{|\alpha |}^{\sigma }}{|\alpha |}^{{\tau |\alpha |}^{\sigma }}$$

hold for all $\alpha \in {\mathbb{N}}_0^d$. For simplicity, assume that $\tau =1$, and the proof for $\tau >1$ is similar.

By the Leibniz formula and $(M.1)$ we have

$$\begin{array}{c}|{\partial }^{\alpha }\left(\varphi \psi \right)\left(x\right)|\le \sum _{\beta \le \alpha }\left(\genfrac{}{}{0pt}{}{\alpha }{\beta }\right)|{\partial }^{\alpha -\beta }\varphi \left(x\right)||{\partial }^{\beta }\psi \left(x\right)|\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le C_{K,h}{\tilde{C}}_{K,k}\sum _{\beta \le \alpha }\left(\genfrac{}{}{0pt}{}{\alpha }{\beta }\right)h^{{|\alpha -\beta |}^{\sigma }}{|\alpha -\beta |}^{{|\alpha -\beta |}^{\sigma }}{k}^{{|\beta |}^{\sigma }}{|\beta |}^{{|\beta |}^{\sigma }}\\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le C_{K,h}{\tilde{C}}_{K,k}{|\alpha |}^{{|\alpha |}^{\sigma }}\sum _{\beta \le \alpha }\left(\genfrac{}{}{0pt}{}{\alpha }{\beta }\right)h^{{|\alpha -\beta |}^{\sigma }}{k}^{{|\beta |}^{\sigma }},x\in K.\end{array}$$

By choosing $h=k$ we obtain

$$\sum _{\beta \le \alpha }\left(\genfrac{}{}{0pt}{}{\alpha }{\beta }\right)h^{{|\alpha -\beta |}^{\sigma }}{k}^{{|\beta |}^{\sigma }}\le 2^{|\alpha |}{h}^{\frac{{|\alpha |}^{\sigma }}{2^{\sigma -1}}}\le {\left(2h^{\frac{1}{2^{\sigma -1}}}\right)}^{{|\alpha |}^{\sigma }},$$

and obtain

$$|{\partial }^{\alpha }\left(\varphi \psi \right)\left(x\right)|\le C{\left(2h^{\frac{1}{2^{\sigma -1}}}\right)}^{{|\alpha |}^{\sigma }}{|\alpha |}^{{|\alpha |}^{\sigma }},$$

with $C=C_{K,h}{\tilde{C}}_{K,h}$.

Thus, for any given $\tilde{h}>0$ we can choose $h<{(\tilde{h}/2)}^{2^{\sigma -1}}$ to obtain

$$|{\partial }^{\alpha }\left(\varphi \psi \right)\left(x\right)|\le C{\tilde{h}}^{{|\alpha |}^{\sigma }}{|\alpha |}^{{|\alpha |}^{\sigma }},$$

where $C>0$ depends on K and $\tilde{h}$, that is, $\varphi \psi \in {\mathcal{E}}_{(\tau ,\sigma )}\left({\mathbb{R}}^d\right)$. □

3.4. Inverse Closedness and Composition

We need some preparation related to the decompositions that appear when using the generalized Faà di Bruno formula.

For multiindices $\alpha ,\beta \in {\mathbb{N}}_0^d$ we write $\alpha \ll \beta$ if there exists $j\in \{1,\cdots ,d\}$ such that ${\alpha }_1={\beta }_1,$$\cdots ,$${\alpha }_{j-1}={\beta }_{j-1}$ and ${\alpha }_j<{\beta }_j.$ We say that $\alpha \in {\mathbb{N}}_0^d$ is decomposed into parts $p_1,\cdots ,p_s\in {\mathbb{N}}_0^d$ with multiplicities $m_1,\cdots ,m_s\in {\mathbb{N}}_0$, if

$$\alpha =m_1{p}_1+m_2{p}_2+\cdots +m_s{p}_s,$$

(22)

where $0\ll p_1\ll p_2\ll \cdots \ll p_s$. Note, that $s\le |\alpha |$ and $m=m_1+\cdots +m_s\le |\alpha |$.

The triple $(s,p,m)$ is called the decomposition of $\alpha \in {\mathbb{N}}_0^d$ and the set of all decompositions of the form (22) is denoted by $\pi$.

For smooth functions $f:\mathbb{R}\to \mathbb{C}$ and $g:{\mathbb{R}}^d\to \mathbb{R}$, the generalized Faà di Bruno formula is given by

$${\partial }^{\alpha }\left(f\left(g\right)\right)=\alpha !\sum _{(s,p,m)\in \pi }{f}^{\left(m\right)}\left(g\right)\prod _{k=1}^s\frac{1}{m_k!}{\left(\frac{1}{p_k!}{\partial }^{p_k}g\right)}^{m_k}.$$

(23)

The total number $\mathrm{card}\pi$ of different decompositions of a multiindex $\alpha \in {\mathbb{N}}_0^d$ given by (22) can be estimated as follows: ${\displaystyle \mathrm{card}\pi \le {(1+|\alpha |)}^{d+2},}$ cf. [28] (Remark 2.2).

Theorem 2. 

Let $\tau >0$, and $\sigma >1$. If $f\in {\mathcal{E}}_{\tau ,\sigma }\left(\mathbb{R}\right)$ and $g\in {\mathcal{E}}_{\tau ,\sigma }\left({\mathbb{R}}^d\right)$ is such that $g:{\mathbb{R}}^d\to \mathbb{R}$, then $f\circ g\in {\mathcal{E}}_{\tau ,\sigma }\left({\mathbb{R}}^d\right)$.

Proof. 

The proof of Theorem 2 for the Roumieu case is given in [7], so we prove the Beurling case here.

Let $K\subset \subset {\mathbb{R}}^d$ be fixed. Put $I=\left\{g\right(x),x\in K\}$ and note that I is a compact set, $I\subset \subset \mathbb{R}$.

For $f\in {\mathcal{E}}_{(\tau ,\sigma )}\left(\mathbb{R}\right)$ and $g\in {\mathcal{E}}_{(\tau ,\sigma )}\left({\mathbb{R}}^d\right)$ by the Faá di Bruno formula we have that for every $h_1,h_2>0$ there are constants $C_{h_1},C_{h_2}>0$ such that

$$\begin{array}{cc}\hfill |{\partial }^{\alpha }(f\circ g)\left(x\right)|& \le |\alpha |!\sum _{(s,p,m)\in \pi }|f^{\left(m\right)}\left(g\left(x\right)\right)|\prod _{k=1}^s\frac{1}{m_k!}{\left(\frac{1}{p_k!}|{\partial }^{p_k}g\left(x\right)|\right)}^{m_k}\hfill \\ \hfill & \le d^{|\alpha |}|\alpha |!\sum _{(s,p,m)\in \pi }\frac{C_{h_2}{h}_2^{m^{\sigma }}{m}^{\tau m^{\sigma }}}{m_1!\cdots m_s!}{C}_{h_1}^m\prod _{k=1}^s{\left(\frac{h_1^{|p_k{|}^{\sigma }}{|p_k|}^{\tau |p_k{|}^{\sigma }}}{|p_k|!}\right)}^{m_k},x\in K,\hfill \end{array}$$

where we used $|p_k|!\le d^{|p_k|}{p}_k!$, $p_k\in {\mathbb{N}}^d$, $k=1,\cdots ,s$.

By choosing $h_2=h_1/2$ and $A_{h_1}=max\{1,C_{h_1},C_{h_2}\}$ we obtain

$$\begin{array}{cc}\hfill |{\partial }^{\alpha }(f\circ g)\left(x\right)|& \le d^{|\alpha |}|\alpha |!{\displaystyle \sum _{\pi }}\frac{h_1^{m^{\sigma }}{m}^{\tau m^{\sigma }}}{m_1!\cdots m_s!}\frac{A_{h_1}^{m+1}}{2^{m^{\sigma }}}{\displaystyle \prod _{k=1}^s}{\left(\frac{h_1^{|p_k{|}^{\sigma }}{|p_k|}^{\tau |p_k{|}^{\sigma }}}{|p_k|!}\right)}^{m_k}\hfill \\ \hfill & \le d^{|\alpha |}|\alpha |!{\displaystyle \sum _{\pi }}\frac{m!}{m_1!\cdots m_s!}{C}_{h_1}^{\prime }\frac{h_1^{m^{\sigma }}{m}^{\tau m^{\sigma }}}{m!}{\displaystyle \prod _{k=1}^s}{\left(\frac{h_1^{|p_k{|}^{\sigma }}{|p_k|}^{\tau |p_k{|}^{\sigma }}}{|p_k|!}\right)}^{m_k},\hfill \end{array}$$

(24)

where ${\displaystyle C_{h_1}^{\prime }:=\underset{n\in {\mathbb{N}}_0}{sup}\frac{A_{h_1}^{n+1}}{2^{n^{\sigma }}}\ge \frac{A_{h_1}^{m+1}}{2^{m^{\sigma }}}}$, ${m\in \mathbb{N}}_0$.

Next, we use [10] (Theorem 5.2 v)) which states that the weight matrix ${\mathcal{M}}^{\tau ,\sigma }=$$\{h^{p^{\sigma }}{p}^{\tau p^{\sigma }}|h>0\}$ satisfies the ${\mathcal{M}}_{\left(\mathrm{FdB}\right)}$ property (cf. [10] (p. 6)). This means that for any $\tilde{h}>0$ there exist $h_1>0$ and $C_{\tilde{h}}^{\prime }>0$ such that

$$\frac{h_1^{m^{\sigma }}{m}^{\tau m^{\sigma }}}{m!}\prod _{k=1}^s{\left(\frac{h_1^{|p_k{|}^{\sigma }}{|p_k|}^{\tau |p_k{|}^{\sigma }}}{|p_k|!}\right)}^{m_k}\le C_{\tilde{h}}^{\prime }{\tilde{h}}^{{|\alpha |}^{\sigma }}\frac{{|\alpha |}^{{\tau |\alpha |}^{\sigma }}}{|\alpha |!}.$$

(25)

Finally, we need the estimate

$$\sum \frac{m!}{m_1!\cdots m_s!}=2^{m_1+2m_2+\cdots +|\alpha |m_{|\alpha |}-1}=2^{|\alpha |-1}\le 2^{{|\alpha |}^{\sigma }},$$

(26)

which is proved in [7] (p. 2770).

By (24)–(26), it follows that for arbitrary $\tilde{h}>0$ we can choose $h_1>0$ such that

$$\underset{x\in K}{sup}|{\partial }^{\alpha }(f\circ g)\left(x\right)|\le C_{\tilde{h}}^{\prime \prime }{\left(2d\tilde{h}\right)}^{{|\alpha |}^{\sigma }}{|\alpha |}^{{\tau |\alpha |}^{\sigma }},$$

where $C_{\tilde{h}}^{\prime \prime }=C_{h_1}^{\prime }{C}_{\tilde{h}}^{\prime }>0$.

Therefore, for any given $h>0$ by choosing $\tilde{h}<h/\left(2d\right)$ we obtain

$$\underset{x\in K}{sup}|{\partial }^{\alpha }(f\circ g)\left(x\right)|\le C_h{h}^{{|\alpha |}^{\sigma }}{|\alpha |}^{{\tau |\alpha |}^{\sigma }},$$

where $C_h:=C_{\tilde{h}}^{\prime \prime }$. Thus $f\circ g\in {\mathcal{E}}_{(\tau ,\sigma )}\left({\mathbb{R}}^d\right)$. □

As a consequence of Theorem 2 we conclude the following:

Corollary 1. 

Let $\tau >0$, and $\sigma >1$. Then, the extended Gevrey class ${\mathcal{E}}_{\tau ,\sigma }\left({\mathbb{R}}^d\right)$ is inverse-closed in $C^{\infty }\left({\mathbb{R}}^d\right)$.

3.5. Paley–Wiener Theorems

Let ${\displaystyle {\mathcal{E}}_{\left(\sigma \right)}\left({\mathbb{R}}^d\right)=\bigcap _{\tau >0}{\mathcal{E}}_{(\tau ,\sigma )}\left({\mathbb{R}}^d\right)}$ and let ${\mathcal{D}}_{\left(\sigma \right)}\left({\mathbb{R}}^d\right)$ denote the set of compactly supported elements from ${\mathcal{E}}_{\left(\sigma \right)}\left({\mathbb{R}}^d\right).$ The dual space of ${\mathcal{E}}_{\left(\sigma \right)}\left({\mathbb{R}}^d\right)$ is given by ${\displaystyle {\mathcal{E}}_{\left(\sigma \right)}^{\prime }\left({\mathbb{R}}^d\right)=\bigcup _{\tau >0}{\mathcal{E}}_{(\tau ,\sigma )}^{\prime }\left({\mathbb{R}}^d\right)}$, and similarly for ${\mathcal{D}}_{\left(\sigma \right)}^{\prime }\left({\mathbb{R}}^d\right)$.

Since we are interested in the intersection of ${\mathcal{E}}_{(\tau ,\sigma )}\left({\mathbb{R}}^d\right)$ with respect to $\tau >0$, we note that if $0<{\tau }_1<{\tau }_2$ and $\sigma >1$ then for every $h>0$ there exists $A>0$ such that

$$A{M}_p^{{\tau }_1,\sigma }\le h^{p^{\sigma }}{M}_p^{{\tau }_2,\sigma },p\in {\mathbb{N}}_0.$$

(27)

This simple observation will be used in the proof of Theorem 3, see also [6].

A more general statement than Proposition 5 is given in [17] (Theorem 3.1).

Proposition 5. 

Let $\sigma >1$, and let $f\in {\mathcal{D}}_{\left(\sigma \right)}\left({\mathbb{R}}^d\right).$ Then, $\widehat{f}$, the Fourier transform of f, is an analytic function, and for every $h>0$ there exists a constant $C_h>0$ such that

$$\begin{array}{c}\hfill |\widehat{f}\left(\xi \right)|\le C_{h}exp\left\{-h\left({ln}^{\frac{\sigma }{\sigma -1}}(|\xi |)/W^{\frac{1}{\sigma -1}}(ln(|\xi |\left)\right)\right)\right\},|\xi |>1,\end{array}$$

(28)

where W denotes the Lambert function.

Proof. 

The analyticity of $\widehat{f}$ follows from the classical Paley–Wiener theorem, cf. [8]. It remains to prove (28).

Let $f\in {\mathcal{D}}_{\left(\sigma \right)}\left({\mathbb{R}}^d\right)$, and let K denote the support of f. Since $f\in {\mathcal{D}}_{\frac{\tau }{2},\sigma }\left({\mathbb{R}}^d\right)$, by Definition 1 for every $\alpha \in {\mathbb{N}}_0^d$ we obtain the following estimate:

$$|{\xi }^{\alpha }\widehat{f}\left(\xi \right)|=|\widehat{D^{\alpha }f}\left(\xi \right)|\le C\underset{x\in K}{sup}|D^{\alpha }f\left(x\right)|\le C_1^{{|\alpha |}^{\sigma }+1}{|\alpha |}^{\frac{\tau }{2}{|\alpha |}^{\sigma }}\le C_2{|\alpha |}^{{\tau |\alpha |}^{\sigma }},\xi \in {\mathbb{R}}^d,$$

for a suitable constant $C_2>0$. Now, the relation between the sequence $\left(M_p^{\tau ,\sigma }\right)$ and its associated function $T_{\tau ,\sigma }$ given by (10) implies that

$$|\widehat{f}\left(\xi \right)|\le C_3\underset{\alpha \in {\mathbb{N}}^d}{inf}\frac{{|\alpha |}^{{\tau |\alpha |}^{\sigma }}}{{|\xi |}^{|\alpha |}}\le C_3{e}^{-T_{\tau ,\sigma }(|\xi |)},\xi \in {\mathbb{R}}^d,$$

for suitable $C_3>0$, where we have also used the inequality ${|\xi |}^{|\alpha |}\le \sqrt{d}|{\xi }^{\alpha }|$, for every $\alpha \in {\mathbb{N}}_0^d$ and $\xi \in {\mathbb{R}}^d$. Then, from the left-hand side of (12) we obtain

$$|\widehat{f}\left(\xi \right)|\le C_{4}exp\left\{-B_{\sigma }{\tau }^{-\frac{1}{\sigma -1}}\left({ln}^{\frac{\sigma }{\sigma -1}}(|\xi |)/W^{\frac{1}{\sigma -1}}(ln(|\xi |\left)\right)\right)\right\},|\xi |>1,$$

with ${\displaystyle C_4=C_3{e}^{-{\tilde{B}}_{\tau ,\sigma }}}$. For any given $h>0$ we choose $\tau ={(B_{\sigma }/h)}^{\sigma -1}$, to obtain (28), which proves the claim. □

We proceed with the Paley–Wiener theorem for $u\in {\mathcal{E}}_{\left(\sigma \right)}^{\prime }\left({\mathbb{R}}^d\right)$.

Theorem 3. 

Let $\sigma >1$.

(i) 

If $u\in {\mathcal{E}}_{\left(\sigma \right)}^{\prime }\left({\mathbb{R}}^d\right)$ then there are constants $h,C>0$ such that

$$|\widehat{u}\left(\xi \right)|\le Cexp\left\{h{\left(\frac{{ln}^{\sigma }|\xi |}{W(ln(|\xi |\left)\right)}\right)}^{\frac{1}{\sigma -1}}\right\},|\xi |>1.$$

(ii) 

If $u\in {\mathcal{E}}_{\left(\sigma \right)}^{\prime }\left({\mathbb{R}}^d\right)$ and if for every $h>0$ there exists $C>0$ such that

$$|\widehat{u}\left(\xi \right)|\le Cexp\left\{-h{\left(\frac{{ln}^{\sigma }(1+|\xi |)}{W(ln(1+|\xi |\left)\right)}\right)}^{\frac{1}{\sigma -1}}\right\},|\xi |\ne 0,$$

(29)

then $u\in {\mathcal{E}}_{\left(\sigma \right)}\left({\mathbb{R}}^d\right)$.

Proof. 

$\left(i\right)$ Fix ${\tau }_0>0$ so that $u\in {\mathcal{E}}_{(2{\tau }_0,\sigma )}^{\prime }\left({\mathbb{R}}^d\right)$. By applying (19) to ${\phi }_{\xi }\left(x\right)=e^{-2\pi ix·\xi }\in {\mathcal{E}}_{\left(\sigma \right)}\left({\mathbb{R}}^d\right)$, $\xi \in {\mathbb{R}}^d$, we obtain

$$\begin{array}{c}|\widehat{u}\left(\xi \right)|=|(u,e^{-2\pi i·\xi })|\le C\underset{\alpha \in {\mathbb{N}}_0^d}{sup}\underset{x\in K}{sup}\frac{|D^{\alpha }\left(e^{-2\pi ix·\xi }\right)|}{{\epsilon }^{{|\alpha |}^{\sigma }}{|\alpha |}^{2{\tau }_0{|\alpha |}^{\sigma }}}\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le C\underset{\alpha \in {\mathbb{N}}_0^d}{sup}\frac{{\left(2\pi \right)}^{|\alpha |}|{\xi }^{\alpha }|}{{\epsilon }^{{|\alpha |}^{\sigma }}{|\alpha |}^{2{\tau }_0{|\alpha |}^{\sigma }}}\le C_1\underset{\alpha \in {\mathbb{N}}_0^d}{sup}\frac{{|\xi |}^{|\alpha |}}{{|\alpha |}^{{\tau }_0{|\alpha |}^{\sigma }}}=C_{1}exp\{T_{{\tau }_0,\sigma }(|\xi |)\},\xi \in {\mathbb{R}}^d,\end{array}$$

where we used $|{\xi }^{\alpha }{|\le |\xi |}^{|\alpha |}$, $\alpha \in {\mathbb{N}}_0^d$, $\xi \in {\mathbb{R}}^d$, and ${(\epsilon /\left(2\pi \right))}^{{|\alpha |}^{\sigma }}{|\alpha |}^{2{\tau }_0{|\alpha |}^{\sigma }}\ge A{|\alpha |}^{{\tau }_0{|\alpha |}^{\sigma }}$ for suitable $A>0$, see (27). Now the statement follows from (12).

$\left(ii\right)$ By (27), it is sufficient to prove that for every $\tau >0$ there exists constant $B>0$ such that

$$\underset{x\in {\mathbb{R}}^d}{sup}|D^{\alpha }{u\left(x\right)|\le B|\alpha |}^{{\tau |\alpha |}^{\sigma }},\alpha \in {\mathbb{N}}_0^d.$$

(30)

We take $A_{\sigma }$ as in (12) and for arbitrary $\tau >0$ let us set $h:=2A_{\sigma }{\tau }^{-\frac{1}{\sigma -1}}$ in (29). Then, the Fourier inversion formula, together with (12) and (29), implies that there are constants $C,C_{\tau ,\sigma }>0$, such that

$$\begin{array}{c}|D^{\alpha }u\left(x\right)|=\left|{\int }_{{\mathbb{R}}^d}{\xi }^{\alpha }\widehat{u}\left(\xi \right)e^{2\pi ix\xi }d\xi \right|\hfill \\ \le C{\int }_{{\mathbb{R}}^d}{|\xi |}^{|\alpha |}exp\left\{-2A_{\sigma }{\tau }^{-\frac{1}{\sigma -1}}{\left(\frac{{ln}^{\sigma }(1+|\xi |)}{W(ln(1+|\xi |\left)\right)}\right)}^{\frac{1}{\sigma -1}}\right\}d\xi \\ \le C_{\tau ,\sigma }\underset{\xi \in {\mathbb{R}}^d}{sup}\left({|\xi |}^{|\alpha |}exp\{-T_{\tau ,\sigma }(1+|\xi |)\}\right){\int }_{{\mathbb{R}}^d}exp\left\{-A_{\sigma }{\left(\frac{{ln}^{\sigma }(1+|\xi |)}{\tau W(ln(1+|\xi |\left)\right)}\right)}^{\frac{1}{\sigma -1}}\right\}d\xi \\ \hfill \le B\underset{\xi \in {\mathbb{R}}^d}{sup}\frac{{|\xi |}^{|\alpha |}}{exp\{T_{\tau ,\sigma }(|\xi |)\}},\xi \in {\mathbb{R}}^d,\end{array}$$

for suitable $B>0$, where the last integral is finite due to the property $(W.3)$ of the Lambert W function. Then, (30) follows from (10) with $p=|\alpha |$. □

4. Wave-Front Sets for Extended Gevrey Regularity

4.1. Wave-Front Set and Singular Support

Wave-front sets measure different types of directional singularities. For example,

$$WF\left(u\right)⊊{WF}_t\left(u\right)⊊{WF}_A\left(u\right),t>1,$$

(31)

where $u\in {\mathcal{D}}^{\prime }\left({\mathbb{R}}^d\right)$, WF is the classical $\left(C^{\infty }\right)$ wave-front set, ${WF}_t$ is the Gevrey wave-front set which corresponds to the Gevrey sequence $\left(p{!}^t\right)$, $t>1$, and ${WF}_A$ is the analytic wave-front set, we refer to [2,8,29] for precise definitions.

In this section, we introduce wave-front sets that detect singularities that are “stronger” then the classical $C^{\infty }$ singularities and “weaker” than any Gevrey singularity. Moreover, the usual properties (such as pseudo-local property), which hold for wave-front sets quoted in (31), are preserved when considering the new type of singularities.

For simplicity, here we consider wave-front sets ${WF}_{\{\tau ,\sigma \}}\left(u\right)$ in terms of extended Gevrey regularity of Roumieu type. Results on ${WF}_{(\tau ,\sigma )}\left(u\right)$ of Beurling type are analogous, cf. [7] (Remark 3.2).

Definition 3. 

Let $u\in {\mathcal{D}}^{\prime }\left({\mathbb{R}}^d\right)$, $\tau >0$, $\sigma >1$, and $(x_0,{\xi }_0)\in {\mathbb{R}}^d\times {\mathbb{R}}^d\setminus \left\{0\right\}$. Then, $(x_0,{\xi }_0)\notin {WF}_{\{\tau ,\sigma \}}\left(u\right)$ if there exists a conic neighborhood ${\Gamma }_0$ of ${\xi }_0$, a compact set $K\subset \subset {\mathbb{R}}^d$, and $\varphi \in {\mathcal{D}}_{\{\tau ,\sigma \}}\left({\mathbb{R}}^d\right)$, $supp\varphi =K$, $\varphi \left(x_0\right)\ne 0$, and such that

$$|\widehat{\varphi u}\left(\xi \right)|\le C\frac{h^{N^{\sigma }}{N}^{\tau N^{\sigma }}}{{|\xi |}^N},N\in {\mathbb{N}}_0,\xi \in {\Gamma }_0,$$

(32)

for some $h>0$ and $C>0$.

Definition 3 does not depend on the choice of the cut-off function $\varphi \in {\mathcal{D}}_{\{\tau ,\sigma \}}\left({\mathbb{R}}^d\right)$ with given properties. We refer to [17] (Theorem 4.2) for the proof of such independence, and note that the inverse closedness property of ${\mathcal{E}}_{\{\tau ,\sigma \}}\left({\mathbb{R}}^d\right)$ (Theorem 1) is used in the proof. Thus, $(x_0,{\xi }_0)\notin {WF}_{\{\tau ,\sigma \}}\left(u\right)$ if (32) holds for all $\varphi \in {\mathcal{D}}_{\{\tau ,\sigma \}}\left({\mathbb{R}}^d\right)$, $supp\varphi =K$, $\varphi \left(x_0\right)\ne 0$, and sometimes it is convenient to assume that $\varphi \equiv 1$ in a neighboorhood of $x_0\in {\mathbb{R}}^d$, cf. [2].

Let $u\in {\mathcal{D}}^{\prime }\left({\mathbb{R}}^d\right)$. Then, ${WF}_{\{\tau ,\sigma \}}\left(u\right)$ is a closed subset of ${\mathbb{R}}^d\times {\mathbb{R}}^d\setminus \left\{0\right\}$, and for every $\tau >0$ and $\sigma >1$ we have

$$WF\left(u\right)⊊{WF}_{\{\tau ,\sigma \}}\left(u\right)⊊{WF}_t\left(u\right)⊊{WF}_A\left(u\right).$$

The singular support of a distribution $u\in {\mathcal{D}}^{\prime }\left({\mathbb{R}}^d\right)$ with respect to extended Gevrey regularity is the complement of the set of points in which u locally belongs to ${\mathcal{E}}_{\{\tau ,\sigma \}}\left({\mathbb{R}}^d\right)$:

Definition 4. 

Let $\tau >0$, $\sigma >1$, and $u\in {\mathcal{D}}^{\prime }\left({\mathbb{R}}^d\right)$. Then, $x_0\notin {singsupp}_{\{\tau ,\sigma \}}\left(u\right)$ if and only if there exists a neighborhood Ω of $x_0$ such that $u\in {\mathcal{E}}_{\{\tau ,\sigma \}}(\Omega )$.

Here, $u\in {\mathcal{E}}_{\{\tau ,\sigma \}}(\Omega )$ means that u satisfies the conditions of Definition 1, i.e., (14), with ${\mathbb{R}}^d$ replaced by its open subset $\Omega$ at each occurrence.

The next result is a consequence of Definitions 3 and 4, we refer to [7] for the proof.

Theorem 4. 

Let $\tau >0$, $\sigma >1$, $u\in {\mathcal{D}}^{\prime }\left({\mathbb{R}}^d\right)$, and let ${\pi }_1:{\mathbb{R}}^d\times {\mathbb{R}}^d\setminus \left\{0\right\}\to {\mathbb{R}}^d$ be the standard projection given by ${\pi }_1(x,\xi )=x$. Then

$${singsupp}_{\{\tau ,\sigma \}}\left(u\right)={\pi }_1\left({WF}_{\{\tau ,\sigma \}}\left(u\right)\right).$$

4.2. Characterization of Wave-Front Sets via the STFT

For the estimates of the short-time Fourier transform it is convenient to consider the following refinement of the associated function $T_{\tau ,\sigma }$ (see (10)).

The two-parameter associated function $T_{\tau ,\sigma }(h,k)$ to the sequence $M_0^{\tau ,\sigma }=1$ and $M_p^{\tau ,\sigma }=p^{\tau p^{\sigma }}$, $p\in \mathbb{N}$, $\tau >0$, $\sigma >1$ is given by

$$T_{\tau ,\sigma }(h,k)=\underset{p\in {\mathbb{N}}_0}{sup}ln\frac{h^{p^{\sigma }}{k}^p}{M_p^{\tau ,\sigma }},h,k>0.$$

(33)

When $h=1$ we recover the associated function with the sequence $\left(M_p^{\tau ,\sigma }\right)$, i.e.,

$$T_{\tau ,\sigma }\left(k\right)=T_{\tau ,\sigma }(1,k),k>0.$$

Sharp asymptotic estimates for $T_{\tau ,\sigma }(h,k)$ in terms of the principal branch of the Lambert function are given in [17].

Here, we recall [21] (Lemma 2), a simple result which relates $T_{\tau ,\sigma }$ and $T_{\tau ,\sigma }(h,·)$, $h>0$.

Lemma 5. 

Let $T_{\tau ,\sigma }(h,k)$ be given by (33), and let $T_{\tau ,\sigma }\left(k\right)$ be given by (10). Then, for any given $h>0$ and ${\tau }_2>\tau >{\tau }_1>0$ there exists $A,B\in \mathbb{R}$ such that

$$T_{{\tau }_2,\sigma }\left(k\right)+A\le T_{\tau ,\sigma }(h,k)\le T_{{\tau }_1,\sigma }\left(k\right)+B,k>0.$$

It is known that the classical wave-front set $WF\left(u\right)$ can be described with the short-time Fourier transform, see [30]. A related characterization of ${WF}_{\{\tau ,\sigma \}}\left(u\right)$ is given in [31]. Here, we provide a slightly different statement and a more detailed proof.

Let there be $f,g\in L^2\left({\mathbb{R}}^d\right).$ The short-time Fourier transform (STFT) of f with respect to the window g is given by

$$V_{g}f(x,\xi )=\int e^{-2\pi it\xi }f\left(t\right)\overline{g(t-x)}dt=\langle f,M_{\xi }{T}_{x}g\rangle ,x,\xi \in {\mathbb{R}}^d.$$

We observe that the definition of $V_{g}f$ makes sense when f and g belong to any dual pairing which extends the inner product in $L^2\left({\mathbb{R}}^d\right)$ as mentioned in Section “Notation”.

We first observe that if $(x_0,{\xi }_0)\notin {WF}_{\{\tau ,\sigma \}}\left(u\right)$ then

$$|\widehat{\varphi u}\left(\xi \right)|\le C\underset{N\in {\mathbb{N}}_0}{inf}\frac{h^{N^{\sigma }}{N}^{\tau N^{\sigma }}}{{|\xi |}^N},\xi \in {\Gamma }_0,$$

(34)

for some $h>0$, $C>0$, and $\varphi$ satisfying the conditions of Definition 3. By (33) (and Lemma 3) it follows that (34) is equivalent to

$$|\widehat{\varphi u}\left(\xi \right)|\le C{e}^{-T_{\tau ,\sigma }(\frac{1}{h},|\xi |)},\xi \in {\Gamma }_0.$$

(35)

Next, we resolve ${WF}_{\{\tau ,\sigma \}}\left(u\right)$ of $u\in {\mathcal{D}}^{\prime }\left({\mathbb{R}}^d\right)$ by considering the asymptotic behavior of its STFT.

In the sequel $\varphi \in {\mathcal{D}}_{\{\tau ,\sigma \}}^K\left({\mathbb{R}}^d\right)$ means that $\varphi \in {\mathcal{D}}_{\{\tau ,\sigma \}}\left({\mathbb{R}}^d\right)$ and $supp\varphi =K$.

Theorem 5. 

Let $u\in {\mathcal{D}}^{\prime }\left({\mathbb{R}}^d\right)$, and $\tau >0$, $\sigma >1$. Then, $(x_0,{\xi }_0)\notin {WF}_{\{\tau ,\sigma \}}\left(u\right)$ if and only if there exists a conic neighborhood ${\Gamma }_0$ of ${\xi }_0$, a compact neighborhood K of $x_0$, and

$$g\in {\mathcal{D}}_{\{\tau ,\sigma \}}^{K_0}\left({\mathbb{R}}^d\right),K_{x_0}=\{y\in {\mathbb{R}}^d|y+x_0\in K\},g\left(0\right)\ne 0,$$

(36)

such that

$$|V_{g}u(x,\xi )|\le C{e}^{-T_{\tau ,\sigma }(k,|\xi |)},x\in K,\xi \in {\Gamma }_0,$$

(37)

for some $k>0$, and $C>0$.

Proof. 

We follow the idea presented in [30], and give the proof to enlighten the difference between $WF\left(u\right)$ and ${WF}_{\{\tau ,\sigma \}}\left(u\right)$.

(⇒) Assume that there is a conic neighborhood $\Gamma$ of ${\xi }_0$, a compact set $K_1$ in ${\mathbb{R}}^d$, so that for any $\varphi \in {\mathcal{D}}_{\{\tau ,\sigma \}}^{K_1}\left({\mathbb{R}}^d\right)$, such that $\varphi \left(x_0\right)\ne 0$, the estimate (35) holds for some $C,h>0$. Without loss of generality, we may assume that $K_1=\overline{B_r\left(x_0\right)}$ for some $r>0$.

By (35) it follows that the set

$$H_h=\{e^{T_{\tau ,\sigma }(\frac{1}{h},|\xi |)}{e}^{-i\xi ·}u(·)|\xi \in {\Gamma }_0\}$$

is weakly bounded, and weakly continuous (since $D_{\{\tau ,\sigma \}}^{K_1}\left({\mathbb{R}}^d\right)$ is barelled).

Let $K=\overline{B_{r/2}\left(x_0\right)}$, and consider the window $g\in {\mathcal{D}}_{\{\tau ,\sigma \}}^{K_{x_0}}\left({\mathbb{R}}^d\right)$, such that $g\ne 0$ on a neighborhood of 0.

Then, $\varphi :=T_x\overline{g}\in {\mathcal{D}}_{\{\tau ,\sigma \}}^{K_1}\left({\mathbb{R}}^d\right)$, and $\varphi \ne 0$ on a neighborhood of $x_0$. By the equicontinuity of $H_h$ it follows that

$$\begin{array}{c}\hfill |\langle e^{T_{\tau ,\sigma }(\frac{1}{h},|\xi |)}{e}^{-i\xi ·}u(·),T_x\overline{g}(·)\rangle |\le C_1\underset{|\alpha |\le N}{sup}\underset{t\in K_1}{sup}|D^{\alpha }g(t-x)|=C_1\underset{|\alpha |\le N}{sup}{\parallel D^{\alpha }g\parallel }_{L^{\infty }}\le C.\end{array}$$

(38)

From the definition of STFT, it follows that

$$V_{g}u(x,\xi )=\widehat{u{T}_x\overline{g}}(x,\xi ),x,\xi \in {\mathbb{R}}^d.$$

This, together with (38) implies

$$|V_{g}u(x,\xi )|=|\widehat{u{T}_x\overline{g}}|\le C{e}^{-T_{\tau ,\sigma }(1/h,|\xi |)},\xi \in \Gamma ,$$

for all $x\in K$, and for some constants $C,h>0$, which gives (37).

Notice that we actually proved that (37) holds for any g satisfying (36).

(⇐) Let the window $g\in {\mathcal{D}}_{\{\tau ,\sigma \}}^{K_{x_0}}\left({\mathbb{R}}^d\right)$, $g\ne 0$ in a neighborhood of 0. Then, $\psi =T_{x_0}\overline{g}\in {\mathcal{D}}_{\{\tau ,\sigma \}}^K\left({\mathbb{R}}^d\right)$, $\psi \left(x_0\right)\ne 0$, and

$$|\widehat{\psi u}\left(\xi \right)|=|V_{g}u(x_0,\xi )|\le A{e}^{-T_{\tau ,\sigma }(k,|\xi |)}\le C\frac{h^{N^{\sigma }}{N}^{\tau N^{\sigma }}}{{|\xi |}^N},N\in {\mathbb{N}}_0,\xi \in {\Gamma }_0,$$

for some $C>0$ and $h=1/k>0$, and the proof is complete. □

By using Lemma 5 and Theorem 1 we can express the decay estimate (37) in terms of the Lambert function as follows.

Corollary 2. 

Let $u\in {\mathcal{D}}^{\prime }\left({\mathbb{R}}^d\right)$, $\tau >0$, $\sigma >1$, and let W denote the Lambert function. If $(x_0,{\xi }_0)\notin {WF}_{\{\tau ,\sigma \}}\left(u\right)$ then there is a conic neighborhood ${\Gamma }_0$ of ${\xi }_0$, a compact neighborhood K of $x_0$, and g satisfying (36) such that

$$|V_{g}u(x,\xi )|\le C{e}^{-c{\left(\frac{{ln}^{\sigma }(|\xi |)}{{\tau }_{2}W(ln(|\xi |\left)\right)}\right)}^{\frac{1}{\sigma -1}}},x\in K,\xi \in {\Gamma }_0,$$

for some $c>0$, $C>0$, and any ${\tau }_2>\tau$.

Conversely, if there exists a conic neighborhood ${\Gamma }_0$ of ${\xi }_0$, a compact neighborhood K of $x_0$, and g satisfying (36) such that

$$|V_{g}u(x,\xi )|\le C{e}^{-c{\left(\frac{{ln}^{\sigma }(|\xi |)}{{\tau }_{1}W(ln(|\xi |\left)\right)}\right)}^{\frac{1}{\sigma -1}}},x\in K,\xi \in {\Gamma }_0,$$

(39)

holds for some $c>0$, $C>0$, and ${\tau }_1<\tau$, then $(x_0,{\xi }_0)\notin {WF}_{\{\tau ,\sigma \}}\left(u\right)$.

As a combination of results from Theorems 3(ii) and 4, we can use Corollary 2 to characterize local regularity of $u\in {\mathcal{D}}^{\prime }\left({\mathbb{R}}^d\right)$. Namely if (39) holds, then $x_0\notin {singsupp}_{\{\tau ,\sigma \}}\left(u\right)$, so that $u\in {\mathcal{E}}_{\{\tau ,\sigma \}}(\Omega )$ in a neighborhood $\Omega$ of $x_0$ (see Definition 4).

4.3. Propagation of Singularities

One of the main properties of wave-front sets is microlocal hypoellipticity. We first recall the notion of the characteristic set of an operator and the main property of its principal symbol.

If $P(x,D)={\sum }_{|\alpha |\le m}{a}_{\alpha }\left(x\right)D^{\alpha }$ is a differential operator of order m in ${\mathbb{R}}^d$ and $a_{\alpha }\in C^{\infty }\left({\mathbb{R}}^d\right)$, $|\alpha |\le m$, then its characteristic set is given by

$$\mathrm{Char}\left(P(x,D)\right)=\bigcup _{x\in {\mathbb{R}}^d}\left\{(x,\xi )\in {\mathbb{R}}^d\times {\mathbb{R}}^d\setminus \left\{0\right\}\mid P_m(x,\xi )=0\right\}.$$

Here, $P_m(x,\xi )={\sum }_{|\alpha |=m}{a}_{\alpha }\left(x\right){\xi }^{\alpha }\in C^{\infty }({\mathbb{R}}^d\times {\mathbb{R}}^d\setminus \left\{0\right\})$ is the principal symbol of $P(x,D).$ If $\mathrm{Char}\left(P\right(x,D\left)\right)=\varnothing$, then the operator $P(x,D)$ is hypoelliptic.

Now, for the Roumieu wave-front ${WF}_{\{\tau ,\sigma \}}\left(u\right)$ we have the following theorem on the propagation of singularities.

Theorem 6. 

Let $\tau >0,$$\sigma >1$, $u\in {\mathcal{D}}^{\prime }\left({\mathbb{R}}^d\right)$ and let $P(x,D)={\sum }_{|\alpha |\le m}{a}_{\alpha }\left(x\right)D^{\alpha }$ be a partial differential operator of order m such that $a_{\alpha }\left(x\right)\in {\mathcal{E}}_{\{\tau ,\sigma \}}\left({\mathbb{R}}^d\right)$, $|\alpha |\le m$. Then

$${WF}_{\{2^{\sigma -1}\tau ,\sigma \}}\left(f\right)\subseteq {WF}_{\{2^{\sigma -1}\tau ,\sigma \}}\left(u\right)\subseteq {WF}_{\{\tau ,\sigma \}}\left(f\right)\cup \mathrm{Char}\left(P(x,D)\right),$$

where $P(x,D)u=f$ in ${\mathcal{D}}^{\prime }\left({\mathbb{R}}^d\right)$. In particular,

$${WF}_{0,\sigma }\left(f\right)\subseteq {WF}_{0,\sigma }\left(u\right)\subseteq {WF}_{0,\sigma }\left(f\right)\cup \mathrm{Char}\left(P(x,D)\right),$$

(40)

where ${WF}_{0,\sigma }\left(u\right)={\bigcap }_{\tau >0}{WF}_{\{\tau ,\sigma \}}\left(u\right)$.

We note that ${WF}_{0,\sigma }\left(f\right)={WF}_{0,\sigma }\left(u\right)$ in (40) reveals the hypoellipticity of $P(x,D)$.

The proof of Theorem 6 given in [7] contains nontrivial modifications of the proof of [8] (Theorem 8.6.1). In addition, although in the formulation of Theorem 6 we assume that $\sigma >1$, by inspection of the proof of [7] (Theorem 4.1) it follows that the same conclusion holds for $\sigma =1$ and $\tau >1$. In such a way we recover the result for propagation of singularities when the coefficients are Gevrey regular functions.

5. Applications

5.1. A Strictly Hyperbolic Partial-Differential Equation

Cicognani and Lorenz in [5] considered the Cauchy problem for strictly hyperbolic m-th order partial-differential equations (PDEs) of the form

$$\begin{array}{c}{D}_t^{m}u={\displaystyle \sum _{j=0}^{m-1}}{A}_{m-j}(t,x,D_x)D_t^{j}u+f(t,x),\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{D}_t^{k-1}u(0,x)=g_k\left(x\right),(t,x)\in [0,T]\times {\mathbb{R}}^d,k=1,\dots ,m,\end{array}$$

(41)

where

$$A_{m-j}(t,x,D_x)=\sum _{|\gamma |+j\le m}{a}_{m-j,\gamma }(t,x)D_x^{\gamma },$$

where f and $g_k$, $k=1,\cdots ,m$, satisfy certain Sobolev type regularity conditions (cf. (SH3-W) and (SH4-W) in [5]), and studied well-posedness when the coefficients are low-regular in time, and smooth in space. More precisely, it is assumed that the coefficients $a_{m-j,\gamma }$ satisfy conditions of the form

$$|D_x^{\beta }{a}_{m-j,\gamma }(t,x)-D_x^{\beta }{a}_{m-j,\gamma }(s,x)|\le C{K}_{|\beta |}\mu (|t-s|),0\le |t-s|\le 1,x\in {\mathbb{R}}^d,$$

(42)

where $\mu$ is a modulus of continuity, and $\left(K_{|\beta |}\right)$ is a defining sequence (also called a weight sequence).

The modulus of continuity $\mu$ is used to describe the (low) regularity in time, whereas $\left(K_{|\beta |}\right)$ describes the regularity in space.

When $\mu$ is a weak modulus of continuity,

$$\mu \left(s\right)=s(log\frac{1}{s}+1){log}^{\left[m\right]}\left(\frac{1}{s}\right),s>1,$$

(Log-Log^[m]-Lip-continuity), a suitable weight function $\eta$ which defines the solution space is chosen to be

$$\eta \left(s\right)=log\left(s\right){\left({log}^{\left[m\right]}\left(s\right)\right)}^{1+\epsilon }+c_m,$$

where $\epsilon >0$ is arbitrarily small and $c_m>0$ such that $\eta \left(s\right)\ge 1$ for all $s>1$.

We refer to [5] for a detailed analysis of (41), and note that the relation between the modulus of continuity $\mu$ and the weight function $\eta$ is given by

$$\underset{|\xi |\to \infty }{lim}\frac{\mu \left(\frac{1}{\langle \xi \rangle }\right)\langle \xi \rangle }{\eta (\langle \xi \rangle )}=0,{\langle \xi \rangle }^2=1+{|\xi |}^2,\xi \in {\mathbb{R}}^d,$$

while the condition that links the weight sequence $\left(K_p\right)$ to the weight function $\eta$ is given by

$$\underset{p\in {\mathbb{N}}_0}{inf}\frac{K_p}{{\langle \xi \rangle }^p}\le C{e}^{-h\eta (\langle \xi \rangle )},$$

for some $h,C>0$, which is essentially the relation between the Carleman-associated function and the Komatsu associated function as given in Lemma 3.

One of the conclusions in [5] is that the Cauchy problem (41) is well-posed if $a_{m-j,\gamma }(t,x)\in {\mathcal{E}}_{\{1,2\}}\left({\mathbb{R}}^d\right)$ uniformly in x for every fixed t. In other words, the sequence $\left(K_{|\beta |}\right)$ in (42) is given by $K_p=p^{p^2},$$p\in {\mathbb{N}}_0$.

5.2. Generalized Definition of Ultradifferentiable Classes

It was recently demonstrated in [10] that the extended Gevrey classes are prominent examples of ultradifferentiable functions defined in the framework of generalized weighted matrices.

The main idea behind the weighted matrices approach given in [24,32] is to establish a general framework for considering the Braun–Meise–Taylor and Komatsu approach to ultradifferentiable functions in a unified way. To include the extended Gevrey classes which are called Pilipović–Teofanov–Tomić classes (PTT-classes) in [10,11], the so-called exponential sequences $\Phi ={\left({\Phi }_p\right)}_{p\in {\mathbb{N}}_0}$, and the related generalized weighted matrix setting are introduced in [10]. One of the main observations in [10] is that the exponential sequences $\Phi$ (such as ${\left(h^{p^{\sigma }}\right)}_{p\in {\mathbb{N}}_0}$, for some $h>0$) yield “ultradifferentiable classes beyond geometric growth factors”, under mild regularity and growth assumptions on $\Phi$. In such context, PTT classes constitute genuine examples of ultradifferentiable functions defined by weight matrices.

This approach reveals that, apart from stability properties mentioned in Section 3, PTT classes enjoy almost analytic extension [33], and almost harmonic extension [34]. Moreover, PTT classes are a convenient tool for the study of Borel mappings. More precisely, the asymptotic Borel mapping, which sends a function into its series of asymptotic expansion in a sector, is known to be surjective for arbitrary openings in the framework of ultraholomorphic classes associated with sequences of rapid growth. By using the PTT-classes ${\mathcal{E}}_{\{\tau ,\sigma \}}\left({\mathbb{R}}^d\right)$, given by $M_p^{\tau ,\sigma }=p^{\tau p^{\sigma }}$, $p\in \mathbb{N}$, $\tau >0$, $1<\sigma <2$, Jiménez-Garrido, Lastra and Sanz, presented a constructive proof of the surjectivity of the Borel map in sectors of the complex plane for the ultraholomorphic class associated with $\left(M_p^{\tau ,\sigma }\right)$. In fact, the asymptotic behavior of the associated function given in terms of the Lambert function (see Theorem 1) plays a prominent role in these investigations. We refer to [11] for more details.

6. Discussion

The family of extended Gevrey classes introduced in [6,7] can be introduced within the generalized weighted matrices approach to ultradifferentiable functions, [10]. Due to the properties of their defining sequences $\left(M_p^{\tau ,\sigma }\right)$, $\tau >0$, $\sigma >1$, the asymptotic behavior of the associated function $T_{\tau ,\sigma }$ can be conveniently described via the Lambert function W. This is an important feature in applications, cf. [5,11]. We believe that these insights will be useful in future investigations, both from a theoretical and applied point of view.