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

TL;DR

This paper proves the continuity of Gevrey-Hörmander pseudo-differential operators on modulation spaces under certain conditions, extending the understanding of their boundedness properties in functional analysis.

Contribution

It establishes the continuity of Gevrey-Hörmander pseudo-differential operators on modulation spaces with weights, broadening the scope of their boundedness results.

Findings

Pseudo-differential operators are continuous between weighted modulation spaces.

The results apply to operators with symbols in Gevrey-Hörmander classes.

The work extends previous boundedness results to a broader class of operators.

Abstract

Let $s\ge 1$, $\omega ,\omega_0\in \mathscr P_{E,s}^0$, $a\in \Gamma _{s}^{(\omega_0)}$, and let $\mathscr B$ be a suitable invariant quasi-Banach function space, Then we prove that the pseudo-differential operator $\operatorname{Op} (a)$ is continuous from $M(\omega_0\omega ,\mathscr B )$ to $M(\omega ,\mathscr B )$.