Decay and smoothness for eigenfunctions of localization operators

We study decay and smoothness properties for eigenfunctions of compact localization operators. Operators with symbols a in the wide modulation space M^{p,∞} (containing the Lebesgue space Lp), p<∞, and windows φ_1,φ_2 in the Schwartz class S are known to be compact. We show that their L^2-eigenfuctions with non-zero eigenvalues are indeed highly compressed onto a few Gabor atoms. Similarly, for symbols a in the weighted modulation spaces , the L^2-eigenfunctions of localization operators are actually Schwartz functions. An important role is played by quasi-Banach Wiener amalgam and modulation spaces. As a tool, new convolution relations for modulation spaces and multiplication relations for Wiener amalgam spaces in the quasi-Banach setting are exhibited.