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

TL;DR

This paper introduces new quasi-Banach modulation spaces on LCA groups, extends Gabor frame theory to quasi-lattices, and applies these tools to analyze boundedness and eigenfunctions of localization operators, recapturing Euclidean results.

Contribution

It develops a novel framework of quasi-Banach modulation spaces on LCA groups and extends Gabor frame theory to quasi-lattices, enabling new boundedness and spectral results for localization operators.

Findings

Established properties of new quasi-Banach modulation spaces.

Extended Gabor frame theory to quasi-lattices.

Proved boundedness and eigenfunction properties of localization operators.

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 quasi-lattices, significantly extending the original theory introduced by Gr\"{o}chenig 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.