Quasi-Banach modulation spaces and localization operators on locally compact abelian groups

Abstract

We introduce new quasi-Banach modulation spaces on locally compact abelian (LCA) groups which coincide with the classical ones in the Banach setting and prove their main properties. Then we study Gabor frames on quasilattices, significantly extending the original theory introduced by Gröchenig and Strohmer. These issues are the key tools in showing boundedness results for Kohn-Nirenberg and localization operators on modulation spaces and studying their eigenfunctions' properties. In particular, the results in the Euclidean space are recaptured.

Comparisons between Fourier and STFT multipliers: the smoothing effect of the Short-time Fourier Transform

“…[10,16]. For a general extension to the quasi-Banach setting on locally compact Abelian groups we mention the recent [8].…”