An L p − L q version of Hardy′s theorem for spherical Fourier transform on semisimple Lie groups

• Published in: International journal of mathematics and mathematical sciences Vol. 2004; no. 33; pp. 1757 - 1769
• Main Authors: Ben Farah, S., Mokni, K., Trimèche, K.
• Format: Journal Article
• Language: English
• Published: 01.01.2004
• ISSN: 0161-1712; 1687-0425
• DOI: 10.1155/S0161171204209140

We consider a real semisimple Lie group G with finite center and K a maximal compact subgroup of G . We prove an L p − L q 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 and ( h a is the heat kernel on G ). We establish that if a b ≥ 1/4 and p or q is finite, then f = 0 almost everywhere. If a b < 1/4, we prove that for all p , q , there are infinitely many nonzero functions f and if a b = 1/4 with p = q = ∞ , we have f = const h a .