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

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 h−1 a f ∈ Lp(G) and eb‖λ‖ (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 = constha.