Anisotropic Gevrey-H\"ormander pseudo-differential operators on modulation spaces

TL;DR

This paper investigates the continuity of anisotropic Gevrey-H"ormander pseudo-differential operators on modulation spaces, establishing boundedness results under specific symbol and weight class conditions.

Contribution

It provides new continuity results for pseudo-differential operators with anisotropic Gevrey-H"ormander symbols on modulation spaces, extending existing theory.

Findings

Operators are continuous from $M( ext{weights})$ to $M( ext{weights})$ under certain symbol classes.

Results apply to invariant Banach function spaces.

Extends the theory of pseudo-differential operators to anisotropic Gevrey-H"ormander classes.

Abstract

We show continuity properties for the pseudo-differential operator $\operatorname{Op} (a)$ from $M(\omega _0\omega ,\mathscr B )$ to $M(\omega ,\mathscr B )$, for fixed $s,\sigma \ge 1$, $\omega ,\omega _0\in \mathscr P _{s,\sigma}^0$ ($\omega ,\omega _0\in \mathscr P _{s,\sigma}$), $a\in \Gamma ^{\sigma,s}_{(\omega _0)}$ ($a\in \Gamma ^{\sigma,s;0}_{(\omega _0)}$) , and $\mathscr B$ is an invariant Banach function space.