Compactness for pseudo-differential and Toeplitz operators on modulation spaces

Abstract

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

48 pages. This is the first version. It is expected that changes will be performed for later versions