Almost periodic pseudodifferential operators and Gevrey classes

Abstract

We study almost periodic pseudodifferential operators acting on almost periodic functions $G_{\rm ap}^s(\rr d)$ of Gevrey regularity index $s\ge 1$. We prove that almost periodic operators with symbols of Hörmander type $S_{\rho ,\delta }^m$ satisfying an $s$-Gevrey condition are continuous on $G_{\rm ap}^s(\rr d)$ provided $0<\rho \le 1$, $\delta =0$ and $s\rho \ge 1$. 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.

40 pages