On Theorems of Hardy, Gelfand–Shilov and Beurling for Semisimple Groups

Abstract

In this paper we prove a strong version of Hardy’s theorem for the group Fourier transform on semisimple Lie groups which characterises the Fourier transforms of all functions satisfying Hardy type conditions. In the particular case of $SL(2,R)$ we characterise all such functions and conjecture that the same is true for all rank one semisimple groups. We also establish an analogue of a theorem of Gelfand and Shilov in the context of semisimple groups. A version of Beurling’s theorem which assumes a Cowling–Price condition on the function is also proved. We show that these results yield most of the earlier results as corollaries.