Compactness properties for modulation spaces

TL;DR

This paper characterizes when embeddings between modulation spaces are compact or continuous, showing that the behavior of weight ratios at infinity determines these properties.

Contribution

It provides necessary and sufficient conditions for the compactness and boundedness of embeddings between modulation spaces based on weight functions.

Findings

Embedding is compact iff the quotient of weights vanishes at infinity.

Embedding is bounded iff the quotient of weights is bounded.

Provides a complete characterization of embedding properties for modulation spaces.

Abstract

We prove that if $\omega _1$ and $\omega _2$ are moderate weights and $\mascB$ is a suitable (quasi-)Banach function space, then a necessary and sufficient condition for the embedding $i\, :\, M (\omega _1,\mascB )\to M (\omega _2,\mascB )$ between two modulation spaces to be compact is that the quotient $\omega _2/\omega _1$ vanishes at infinity. Moreover we show, that the boundedness of $\omega _2/\omega _1$ a necessary and sufficient condition for the previous embedding to be continuous.