Extended Gevrey regularity via the short-time Fourier transform

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.