Multilinear estimates for the Laplace spectral projectors on compact manifolds.
type: Item
instance of: scholarly article
title: Multilinear estimates for the Laplace spectral projectors on compact manifolds.
zbMATH Open document ID: 1040.58011
author: Nicolas Burq
author: Patrick Gérard
author: Nikolay Tzvetkov
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2004-03-15T00:00:00Z
review text: Let \(\Delta\) be the scalar Laplacian on a closed \(m\) dimensional Riemannian manifold. Denote by \(\chi_\lambda:=\chi(\sqrt{-\Delta}-\lambda)\) the spectral projector around \(\lambda\). The authors show there is a constant \(C\) so that for \(\lambda,\mu\geq1\) and \(f,g\in L^2(M)\), one has the estimate \[ \| \chi_\lambda f\chi_\mu g\| _{L^2}\leq C\Lambda(m,\min(\lambda,\mu))\| f\| _{L^2} \| g\| _{L^2} \] where \(\Lambda(2,\nu)=\nu^{1/4}\), \(\Lambda(3,\nu)=\nu^{1/2}\log^{1/2}(\nu)\), and \(\Lambda(m,\nu)=\nu^{(m-2)/2}\) for \(m\geq4\). The authors also establish a corresponding trilinear estimate. Applying this to spherical harmonics \(H_p\) and \(H_q\) of degrees \(p\geq1\) and \(q\geq1\) on \(S^m\) then yields \[ \| H_pH_q\| _{L^2}\leq C\Lambda(m,\min(p,q))\| H_p\| _{L^2}\| H_q\| _{L^2}. \]
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 33C55
zbMATH DE Number: 2057114
zbMATH Keywords: spectral projection
zbMATH Keywords: bilinear estimate
zbMATH Keywords: trilinear estimate
zbMATH Keywords: spherical harmonics
reviewed by: Peter B. Gilkey
MaRDI profile type: Publication
full work available at URL: j.crma.2003.12.015
OpenAlex ID: W1994976747
cites work: Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces
cites work: Riemannian geometry
cites work: Oscillatory integrals and multipliers on FL\(^p\)
cites work: Oscillatory integrals and spherical harmonics
cites work: Concerning the \(L^ p\) norm of spectral clusters for second-order elliptic operators on compact manifolds
cites work: Q4693724
cites work: Q3142876
DOI: 10.1016/J.CRMA.2003.12.015
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description: scientific article; zbMATH DE number 2057114