Compactness for pseudo-differential and Toeplitz operators on modulation spaces

TL;DR

This paper establishes norm equivalences and convolution estimates for a specific modulation space, and applies these to analyze the compactness of pseudo-differential operators on modulation spaces.

Contribution

It introduces a new modulation space $M^{lat ,q}_{( ext{omega})}$, characterizes its completion, and applies this to compactness results for pseudo-differential operators.

Findings

Proves norm equivalences for $M^{lat ,q}_{( ext{omega})}$.

Shows $M^{lat ,q}_{( ext{omega})}$ is the completion of $ ext{Sigma}_1$.

Establishes compactness criteria for pseudo-differential operators with symbols in $M^{lat ,q}_{( ext{omega})}$.

Abstract

We deduce various norm equivalences, and convolution estimates for the modulation space $M^{\sharp ,q}_{(\omega )}$ consisting of all $f\in M^{\infty ,q}_{(\omega )}$ such that $|V_\phi f \cdot \omega |$ satisfies a mild vanishing condition at infinity. We prove that $M^{\sharp ,q}_{(\omega )}$ is the completion of the Gelfand-Shilov space $\Sigma _1$ under the $M^{\infty ,q}_{(\omega )}$ norm. We use these results to deduce compactness for $\Psi$DO $\op (\mathfrak a )$, with $\mathfrak a \in M^{\sharp ,q}_{(\omega )}$, $0<q\le 1$, when acting on a broad family of modulation spaces.