On the action of pseudo-differential operators on Gevrey spaces

TL;DR

This paper investigates how pseudo-differential operators affect Gevrey spaces, introducing symbol classes with spatial Gevrey regularity and analyzing regularity loss through para-product decomposition.

Contribution

It introduces new classes of symbols with spatial Gevrey regularity and analyzes the regularity loss induced by pseudo-differential operators.

Findings

Characterization of symbol classes with spatial Gevrey regularity

Analysis of regularity loss depending on frequency

Application of para-product decomposition to study operator action

Abstract

In this paper we study the action of pseudo-differential operators acting on Gevrey spaces. We introduce classes of classical symbols with spatial Gevrey regularity. As the spatial Gevrey regularity of a symbol $p(\cdot,\xi)$ may depend on the frequency $\xi$, the action of the associated pseudo-differential operator $op(p)$ may induce a loss of regularity. The proof is based on a para-product decomposition.