Wiener tauberian theorems for SL₂(R)

Rudra P. Sarkar

Vol. 177 (1997), No. 2, 291–304

DOI: 10.2140/pjm.1997.177.291

Abstract

In this article we prove a Wiener Tauberian theorem for L^p(SL₂(ℝ)), 1 ≤ p < 2. Let G be the group SL₂(ℝ) and K its maximal compact subgroup SO(2, ℝ). Let M be {±I}. We show that if the Fourier transforms of a set of functions in L^p(G) do not vanish simultaneously on any irreducible L^p−𝜖-tempered representation for some 𝜖 > 0, where they are assumed to be defined, and if for each M-type at least one of the matrix coefficients of any of those Fourier transforms does not ‘decay too rapidly at ∞’ in a certain sense, then this set of functions generate L^p(G) as a L¹(G)-bimodule. As a key step towards this main theorem we prove a W-T Theorem for L^p-sections of certain line bundles over G∕K. W-T theorems on SL₂(ℝ) have been proved so far, for biinvariant L¹ functions and for L¹ functions on the symmetric space SL₂(ℝ)∕SO(2, ℝ), where the generator is left K-finite. Our results are on the space of all L^p functions (resp. sections), p ∈ [1,2) of SL₂(ℝ) (resp. of line bundles over SL₂(ℝ)∕SO(2, ℝ)), without any restriction of K-finiteness on the generators.

Milestones

Received: 30 January 1995

Revised: 7 September 1995

Published: 1 February 1997