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

S. Ben Farah, K. Mokni and K. Trimèche

International Journal of Mathematics and Mathematical Sciences, 2004, vol. 2004, 1-13

Abstract:

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 h a − 1 f ∈ L p ( G ) and e b ‖ λ ‖ 2 ℱ ( f ) ∈ L q ( 𝔞 + * ) ( 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 .

Date: 2004
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:525981

DOI: 10.1155/S0161171204209140

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