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

Abstract

We deduce one-parameter group properties for pseudo-differential operators $\operatorname{Op} (a)$, where $a$ belongs to the class $Γ^{(ω_0)}_*$ 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 $a\in Γ^{(ω_0)}_*$ and $b\in Γ^{(1/ω_0)}_*$. We apply these results to deduce lifting property for modulation spaces and construct explicit isomorpisms between them. For each weight functions $ω,ω_0$ moderated by GRS submultiplicative weights, we prove that the Toeplitz operator (or localization operator) $\operatorname{Tp} (ω_0)$ is an isomorphism from $M^{p,q}_{(ω)}$ onto $M^{p,q}_{(ω/ω_0)}$ for every $p,q \in (0,\infty ]$.