Decay and Smoothness for Eigenfunctions of Localization Operators

Federico Bastianoni, Elena Cordero, Fabio Nicola

Published 2019-02-09Version 1

We study decay and smoothness properties for eigenfunctions of localization operators. Considering symbols in the wide modulation space M^{p,\infty}(R^{2d}) (containing the Lebesgue space L^p(R^{2d})), p < \infty, and two general windows in the Schwartz class S(R^d), we show that L^2-eigenfuctions with non-zero eigenvalue are indeed highly compressed onto a few Gabor atoms. Similarly, for symbols in the weighted modulation space M^\infty_{v_s\otimes 1}(R^{2d}), s > 0, the corresponding L^2-eigenfunctions are actually in S(R^d). 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.