A genuine analogue of the Wiener Tauberian theorem for some Lorentz spaces on \(\mathrm{SL}(2, \mathbb{R} )\)
type: Item
instance of: scholarly article
title: A genuine analogue of the Wiener Tauberian theorem for some Lorentz spaces on \(\mathrm{SL}(2, \mathbb{R} )\)
zbMATH Open document ID: 1468.43004
author: Tapendu Rana
published in: Forum Mathematicum
publication date: 2021-03-09T00:00:00Z
review text: The paper provides a genuine analogue of the Wiener Tauberian theorem for certain Lorentz spaces \(L^{1,p}(G)\), \(1 \leq p < 2\), where \(G\) is the group \(\mathrm{SL}(2,\mathbb{R})\); that is, an ideal generated by \(\Lambda\subseteq L^{1,p}(G)\) shall be dense in \(L^{1,p}(G)\) if the Fourier transforms \(\{\hat{f}: f \in \Lambda\}\) admit a certain non-vanishing condition and satisfy a so-called ``not decay too rapidly'' requirement.\par The proof of the theorem consists in first showing the theorem on suitably well-chosen subsets of \(L^{1,p}(G)\), where a dense set can be constructed as the span of certain eigenfunctions of the Casimir operator. The theorem on these subsets is proven by combining this construction with ideas on the resolvent transform. The theorem in general is then shown by techniques inspired by \textit{R.~Sarkar} [Pac. J. Math. 177, 291--304 (1997; Zbl 0868.43010)].
reviewed by: Gregory Debruyne
Mathematics Subject Classification ID: 43A80
Mathematics Subject Classification ID: 22E30
zbMATH DE Number: 7318800
zbMATH Keywords: resolvent transform
zbMATH Keywords: Wiener Tauberian theorem
zbMATH Keywords: estimate of hypergeometric functions
MaRDI profile type: Publication
full work available at URL: forum-2020-0134
OpenAlex ID: W3106560646
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DOI: 10.1515/FORUM-2020-0134
description: scientific article; zbMATH DE number 7318800