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 $\Gamma ^{(\omega 0)}$ 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 $p,q \in (0,\infty ]$.