Wiener Tauberian theorem for rank one semisimple Lie groups and for hypergeometric transforms

Abstract

We prove a genuine analogue of the Wiener Tauberian theorem for L1false(G//Kfalse), where G is a real rank one noncompact, connected, semisimple Lie group with finite centre. This generalizes the corresponding result on the automorphism group of the unit disk by Y. Ben Natan, Y. Benyamini, H. Hedenmalm, and Y. Weit. We extend this result for hypergeometric transforms and as an application we prove an analogue of Furstenberg theorem on harmonic functions for hypergeometric transforms.

Wiener Tauberian Theorems for Certain Banach Algebras on Real Rank One Semisimple Lie Groups

“…Now sending r → 0 and using the asymptotic behaviour of Φ λ (a t ) near t = 0, we get (see [21,Lemma 8.1])…”

“…Similarly, putting f = Φ λ , g = φ λ in Equation (3-7), we get [φ λ , Φ λ ](•) is constant on (0, ∞). Next, using the asymptotic behaviour of Δ(t) and Φ λ (a t ) near t = ∞, we get (see [21,Lemma 8.1])…”

“…Overview of the proof. We mention that to prove our main results, we follow the approach in [2,21], which uses the resolvent transform method. The outline of the proof of Theorem 1.2 is as follows.…”

A genuine analogue of Wiener Tauberian theorem for $ \mathrm {SL}(2, \R)$

“…The last line of the inequalities follows from similar calculation of [PS,Lemma 3.3], which uses (2.10). Hence the proof follows.…”

“…By the estimates of ||b λ || 1 , ||T λ f || 1 and using a continuity argument we get the necessary estimate of R[g](λ). Then using a log-log type theorem [PS,Theorem 6.3] we show R[g] = 0. 6.…”