An Lp−Lq version of Hardy's theorem for spherical Fourier transform on semisimple Lie groups

Abstract

We consider a real semisimple Lie group G with finite center and K a maximal compact subgroup of G. We prove an Lp−Lq version of Hardy's theorem for the spherical Fourier transform on G. More precisely, let a, b be positive real numbers, 1≤p, q≤∞, and f a K-bi-invariant measurable function on G such that ha−1f∈Lp(G) and eb‖λ‖2ℱ(f)∈Lq(𝔞+*) (ha is the heat kernel on G). We establish that if ab≥1/4 and p or q is finite, then f=0 almost everywhere. If ab<1/4, we prove that for all p, q, there are infinitely many nonzero functions f and if ab=1/4 with p=q=∞, we have f=const ha.