Asymmetry of convolution norms on Lie groups
type: Item
instance of: scholarly article
title: Asymmetry of convolution norms on Lie groups
zbMATH Open document ID: 0961.43005
author: Sanjiv Gupta
author: Fulvio Ricci
author: Anthony H. Dooley
published in: Journal of Functional Analysis
publication date: 2001-05-20T00:00:00Z
review text: In the case of locally compact abelian groups if a convolution operator on \(L^p\) for some \(p>1\) is bounded, then it is also bounded on the dual space \(L^q\) and has the same norm. The present authors show the situation to be completely different for all nonabelian Lie groups except those locally isomorphic to \(R^3\) or to the semidirect product \(R^+\times R^2\) where the techniques do not apply. In particular the authors show that for all Lie groups of the above type there is a convolution operator with kernel supported on an arbitrary neighborhood of the identity and, for a given \(p>1\) and \(p\) not equal to 2, it is bounded on \(L^p\) but unbounded on \(L^q\) for every \(q\) outside the closed interval with endpoints \(p\) and 2.
Mathematics Subject Classification ID: 43A22
Mathematics Subject Classification ID: 22E30
zbMATH DE Number: 1496046
zbMATH Keywords: convolution operator
zbMATH Keywords: nonabelian Lie groups
MaRDI profile type: Publication
full work available at URL: jfan.2000.3573
OpenAlex ID: W2071484922
cites work: The Fourier-Stieltjes algebra of a semisimple group
cites work: Transferring \(L^ p\) multipliers
cites work: On \(L^ p\) multipliers of Cartan motion groups
cites work: On norms of trigonometric polynomials on \(SU(2)\)
cites work: The contraction of K to \=NM
cites work: Singular integrals associated to the Laplacian on the affine group \(ax+b\)
cites work: Harmonic synthesis for subgroups
cites work: Une généralisation de la notion de transformée de Fourier-Stieltjes
cites work: On the asymmetry of norms of convolution operators. I
cites work: Q4164380
cites work: Estimations \(L^ p\) des coefficients de représentation et opérateurs de convolution
cites work: Q5186146
cites work: Msub(p)(G) non= Msub(q)(G) (psup(-1)+qsup(-1)=1)
cites work: Q4118027
cites work: Transferring Fourier Multipliers from SU(2) to the Heisenberg Group
cites work: Harmonic analysis on the group of rigid motions of the Euclidean plane
DOI: 10.1006/JFAN.2000.3573
description: scientific article; zbMATH DE number 1496046