An \(L^p\)-\(L^q\) version of Hardy's theorem for spherical Fourier transform on semisimple Lie groups
type: Item
instance of: scholarly article
title: An \(L^p\)-\(L^q\) version of Hardy's theorem for spherical Fourier transform on semisimple Lie groups
zbMATH Open document ID: 1122.43002
DOI: 10.1155/S0161171204209140
author: Khalifa Trimèche
author: Slaim Ben Farah
author: Kamel Mokni
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2005-05-18T00:00:00Z
full work available at URL: https://eudml.org/doc/53506
review text: Summary: We consider a real semisimple Lie group \(G\) with finite center and \(K\) a maximal compact subgroup of \(G\). We prove an \(L^p-L^q\) version of Hardy's theorem for the spherical Fourier transform on \(G\). More precisely, let \(a\), \(b\) be positive real numbers, \(1\leq p\), \(q\leq\infty\), and \(f\) a \(K\)-bi-invariant measurable function on \(G\) such that \(h_a^{-1}f\in L^p(G)\) and \(e^{b\|\lambda\|^2}\mathcal{F}(f)\in L^q({\mathfrak{a}_+^*})\) (\(h_a\) is the heat kernel on \(G\)). We establish that if \(ab\geq1/4\) and \(p\) or \(q\) is finite, then \(f=0\) almost everywhere. If \(ab<1/4\), we prove that for all \(p\), \(q\), there are infinitely many nonzero functions \(f\) and if \(ab=1/4\) with \(p=q=\infty\), we have \(f=\text{const} h_a\).
Mathematics Subject Classification ID: 43A30
Mathematics Subject Classification ID: 22E30
zbMATH DE Number: 2168724
Wikidata QID: Q115243332
MaRDI profile type: Publication
OpenAlex ID: W2118191786
description: scientific article; zbMATH DE number 2168724