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

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.