Time-frequency representations on Lorentz spaces over locally compact Abelian groups

Abstract

Let $G$ be a locally compact Abelian group with a fixed Haar measure and, denote by $\widehat{G}$ its dual group. In this article, the authors obtain various boundedness of the short-time Fourier transform on Lorentz spaces: $$L^{p_1,u}(G)\times L^{p_2,v}(G)\to L^{q,w}(G\times\widehat{G})$$ with the indexes satisfying appropriate relations. These results are then used to prove the corresponding boundedness of $\tau$-Wigner transforms and $\tau$-Weyl operators. As an application, the Lieb's uncertainty principle in the context of Lorentz spaces is finally investigated. All these results are new even for the case when $G$ is finite.