Compactness 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 $i\, :\, M (ω_1,\mascB )\to M (ω_2,\mascB )$ 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.

25 pages