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 $\mathrm{Op}(a)$, where $a$ belongs to the class ${\Gamma }_{\ast }^{({\omega }_0)}$ of certain Gevrey symbols. We use this to show that there are pseudo-differential operators $\mathrm{Op}(a)$ and $\mathrm{Op}(b)$ which are inverses to each others, where $a\in {\Gamma }_{\ast }^{({\omega }_0)}$ and $b\in {\Gamma }_{\ast }^{(1/{\omega }_0)}$. 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) $\mathrm{Tp}({\omega }_0)$ is an isomorphism from $M_{(\omega )}^{p,q}$ onto $M_{(\omega /{\omega }_0)}^{p,q}$ for every $p,q\in (0,\infty ]$.

43 pages. This is the first version. It is expected that it will be some major changes in the future. arXiv admin note: text overlap with arXiv:0905.4954