Spectrum of the Laplacian and Riesz transform on locally symmetric spaces
type: Item
instance of: scholarly article
title: Spectrum of the Laplacian and Riesz transform on locally symmetric spaces
zbMATH Open document ID: 1166.58015
author: Nikolaos Mandouvalos
author: Michel Marias
published in: Bulletin des Sciences Mathématiques
publication date: 2009-03-11T00:00:00Z
review text: Let \(M\) be a complete non-compact connected Riemannian manifold. Let \(|\omega|_{L^p(\Lambda^q)}\) be the \(L^p\) norm of a \(q\)-form \(\omega\). Let \(d\) be the usual exterior derivative. The square root \(\Delta^{1/2}\) of the Laplace-Beltrami operator is characterized by the identity \(|\Delta^{1/2}f|_{L^2(\Lambda^0)}:=|df|_{L^2(\Lambda^1)}\). One says that the Riesz transformation is bounded on \(L^p\) if there exists a constant \(c_p\) so that one has the estimate \(|df|_{L^p(\Lambda^1)}\leq c_p|\Delta^{1/2}f|_{L^p(\Lambda^0)}\) for all \(f\) with compact support on \(M\). Assume that the discrete part of the spectrum of \(\Delta\) is non-trivial and that \(M\) is a local symmetric space. The authors show that the Riesz transform is bounded on \(L^p\) for all \(p\) in some open neighborhood of \(2\).
reviewed by: Peter B. Gilkey
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 22E30
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 42B20
zbMATH DE Number: 5527922
zbMATH Keywords: local symmetric space
zbMATH Keywords: Kleinian group
zbMATH Keywords: Riesz transform
zbMATH Keywords: Laplace-Beltrami operator
MaRDI profile type: Publication
full work available at URL: j.bulsci.2008.04.005
OpenAlex ID: W2161500631
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DOI: 10.1016/J.BULSCI.2008.04.005
description: scientific article; zbMATH DE number 5527922