© Hindawi Publishing Corp. 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−1a f ∈ Lp(G) and eb‖λ‖2(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. 2000 Mathematics Subject Classification: 22E30, 22E46, 43A30