Wiener Tauberian Theorem for rank one symmetric spaces

Rudra P. Sarkar

Vol. 186 (1998), No. 2, 349–358

DOI: 10.2140/pjm.1998.186.349

Abstract

In this article we prove a Wiener Tauberian (W-T) theorem for L^p(G∕K), p ∈ [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.

Milestones

Received: 21 August 1996

Revised: 8 December 1997

Published: 1 December 1998