Abstract

In this article we prove a Wiener Tauberian (W-T) theorem for L^p(G ∕ K), p in [1,2), where G is one of the semisimple Lie groups of real rank one, SU(n,1),SO(n,1),Sp(n,1) or the connected Lie group of real type F₄,and K is its maximal compact subgroup. W-T theorem for noncompact symmetric space has been proved so far for L¹(SL₂(R) ∕ SO₂(R)) where the generator is necessarily K-finite ([A. Sitaram and M. Sundari, An analogue of Hardy's theorem for very rapidly decreasing functions on semi-simple Lie groups, Pacific J. of Math., 177 (1997), 187–200]). We generalize that result to the case of L^p functions of real rank one groups, without any K-finiteness restriction on the generator. We also obtain a reformulation of the W-T theorems using Hardy’s theorem for semisimple Lie groups.

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