Logarithmic Harnack inequalities
type: Item
instance of: scholarly article
title: Logarithmic Harnack inequalities
zbMATH Open document ID: 0880.58026
published in: Mathematical Research Letters
publication date: 1998-01-08T00:00:00Z
review text: Let \(M\) be a \(d-\)dimensional smooth connected compact Riemannian manifold with non-negative Ricci curvature and let \(\nabla\) denote the gradient associated with the Laplace-Beltrami operator \(\Delta.\) The authors prove the following logarithmic Harnack inequality: \[ |\nabla f|^2+\alpha f^2 \log f^2 \leq \alpha \sup(f^2\log f^2) \] for \(f\) satisfying \[ \Delta f=-\alpha f \log f^2.\tag \(*\) \] Further, equation \((*)\) is satisfied by the function \(f,\) achieving the smallest constant \(\alpha\) (log-Sobolev constant) in the logarithmic Sobolev inequality: \[ \int_M f^2(x)\log f^2(x) \leq \alpha \int_M |\nabla f(x)|^2 , \qquad \int_M |f|^2 =\text{vol } M. \] In addition, \(\alpha \geq \min \{\lambda_1/8e, d^{-1} D(M)^{-2}\},\) where \(\lambda_1\) is the first eigenvalue of the Laplacian and \(D(M)\) denotes the diameter of \(M.\) Similar results are proved for some graphs.
reviewed by: Georgi E. Karadzhov
Mathematics Subject Classification ID: 58J50
zbMATH DE Number: 991916
zbMATH Keywords: Riemannian manifold
zbMATH Keywords: eigenvalues of the Laplacian
zbMATH Keywords: logarithmic Sobolev inequality
author: Fan Chung
author: Shing-Tung Yau
MaRDI profile type: Publication
full work available at URL: mrl.1996.v3.n6.a8
OpenAlex ID: W2334471843
DOI: 10.4310/MRL.1996.V3.N6.A8
description: scientific article; zbMATH DE number 991916