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International Journal of Mathematics and Mathematical Sciences
Volume 2004, Issue 33, Pages 1757-1769
http://dx.doi.org/10.1155/S0161171204209140

An LpLq version of Hardy's theorem for spherical Fourier transform on semisimple Lie groups

1Département de Mathématiques, Faculté des Sciences de Monastir, Monastir 5019, Tunisia
2Département de Mathématiques, Faculté des Sciences de Tunis, Tunis 1060, Tunisia

Received 17 September 2002

Copyright © 2004 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider a real semisimple Lie group G with finite center and K a maximal compact subgroup of G. We prove an LpLq version of Hardy's theorem for the spherical Fourier transform on G. More precisely, let a, b be positive real numbers, 1p, q, and f a K-bi-invariant measurable function on G such that ha1fLp(G) and ebλ2(f)Lq(𝔞+*) (ha is the heat kernel on G). We establish that if ab1/4 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.