A Paley–Wiener theorem in extended Gevrey regularity

Abstract

In this paper we introduce appropriate associated function to the sequence $$M_p=p^{\tau p^{\sigma }}, p\in {\mathbf {N}}, \tau>0, \sigma >1$$, and derive its sharp asymptotic estimates in terms of the Lambert W function. These estimates are used to prove a Paley–Wiener type theorem for compactly supported functions from extended Gevrey classes. As an application, we discuss properties of the corresponding wave front sets.