A genuine analogue of the Wiener Tauberian theorem for some Lorentz spaces on SL(2,ℝ)

Abstract

In this paper, we prove a genuine analogue of the Wiener Tauberian theorem for {L^{p,1}(G)} ({1\leq p<2}), with {G=\mathrm{SL}(2,\mathbb{R})}.

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

“…It is of interest to know whether the Wiener Tauberian theorem holds for the spaces above. The author in [22] answered this affirmatively by proving an analogue of the Wiener Tauberian theorem for L p,1 (SL(2, R)) (1 ≤ p < 2). Our next result is a Wiener Tauberian theorem for L p,1 (G//K) (1 ≤ p < 2), where G is a complex semisimple Lie group of real rank one; that is, G = SL(2, C).…”

“…(4-4) [22] The integral in Equation (4-4) converges since 2/(Imλ + 1)p < 1 for Imλ > γ p . Next, from Equations (4-3) and (4-4),…”