# Extended Gevrey regularity via the short-time Fourier transform

## TL;DR

This paper characterizes the regularity of functions with derivatives controlled by specific sequences using the decay of their short-time Fourier transforms, extending Gevrey regularity results and introducing a new wave front set concept.

## Contribution

It extends Gevrey regularity analysis via the STFT, relaxes support assumptions, and introduces a new wave front set for extended Gevrey classes.

## Key findings

- Characterization of regularity through STFT decay
- A Paley-Wiener type theorem for extended Gevrey classes
- Introduction of a new wave front set concept

## Abstract

We study the regularity of smooth functions whose derivatives are dominated by sequences of the form $M_p^{\tau,\s}=p^{\tau p^{\s}}$, $\tau>0$, $\s\geq1$. We show that such functions can be characterized through the decay properties of their short-time Fourier transforms (STFT), and recover \cite[Theorem 3.1]{CNR} as the special case when $ \t>1$ and $\s = 1$, i.e. when the Gevrey type regularity is considered. These estimates lead to a Paley-Wiener type theorem for extended Gevrey classes. In contrast to the related result from \cite{PTT-05, PTT-04}, here we relax the assumption on compact support of the observed functions. Moreover, we introduce the corresponding wave front set, recover it in terms of the STFT, and discuss local regularity in such context.