Liftings for ultra-modulation spaces, and one-parameter groups of Gevrey type pseudo-differential operators

TL;DR

This paper explores the properties of Gevrey class pseudo-differential operators, establishing their group behavior, invertibility, and applications to lifting properties and isomorphisms in modulation spaces.

Contribution

It introduces one-parameter group properties for Gevrey pseudo-differential operators and applies these to demonstrate isomorphisms and lifting properties in modulation spaces.

Findings

Existence of inverse pseudo-differential operators within Gevrey classes.

Isomorphism of Toeplitz operators between weighted modulation spaces.

Lifting property for modulation spaces via pseudo-differential operators.

Abstract

We deduce one-parameter group properties for pseudo-differential operators $\operatorname{Op} (a)$, where $a$ belongs to the class $\Gamma ^{(\omega _0)}_*$ of certain Gevrey symbols. We use this to show that there are pseudo-differential operators $\operatorname{Op} (a)$ and $\operatorname{Op} (b)$ which are inverses to each others, where $a\in \Gamma ^{(\omega _0)}_*$ and $b\in \Gamma ^{(1/\omega _0)}_*$. We apply these results to deduce lifting property for modulation spaces and construct explicit isomorpisms between them. For each weight functions $\omega ,\omega _0$ moderated by GRS submultiplicative weights, we prove that the Toeplitz operator (or localization operator) $\operatorname{Tp} (\omega _0)$ is an isomorphism from $M^{p,q}_{(\omega )}$ onto $M^{p,q}_{(\omega /\omega _0)}$ for every $p,q \in (0,\infty ]$.