Almost periodic pseudodifferential operators and Gevrey classes

TL;DR

This paper investigates the continuity and regularity properties of almost periodic pseudodifferential operators with Gevrey regularity, establishing a calculus and applying it to hypoelliptic operators.

Contribution

It develops a calculus for almost periodic pseudodifferential operators with Gevrey symbols and proves their continuity and regularity properties on Gevrey almost periodic function spaces.

Findings

Operators are continuous on Gevrey almost periodic functions under certain conditions.

A calculus for symbols and operators is constructed using a regularizing operator.

Regularity results are obtained for a class of hypoelliptic operators.

Abstract

We study almost periodic pseudodifferential operators acting on almost periodic functions $G_{\rm ap}^s(\rr d)$ of Gevrey regularity index $s \geq 1$. We prove that almost periodic operators with symbols of H\"ormander type $S_{\rho,\delta}^m$ satisfying an $s$-Gevrey condition are continuous on $G_{\rm ap}^s(\rr d)$ provided $0 < \rho \leq 1$, $\delta=0$ and $s \rho \geq 1$. A calculus is developed for symbols and operators using a notion of regularizing operator adapted to almost periodic Gevrey functions and its duality. We apply the results to show a regularity result in this context for a class of hypoelliptic operators.