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

We introduce new quasi-Banach modulation spaces on locally compact abelian 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 Grochenig 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.