Remarks on logarithmic Sobolev constants, exponential integrability and bounds on the diameter
type: Item
instance of: scholarly article
title: Remarks on logarithmic Sobolev constants, exponential integrability and bounds on the diameter
zbMATH Open document ID: 0836.60074
published in: Journal of Mathematics of Kyoto University
publication date: 1996-02-12T00:00:00Z
review text: Let \(\rho_0\) be the logarithmic Sobolev constant of the Laplacian on a Riemannian manifold \(M\) with finite (normalized) Riemannian measure \(\mu\). Using a simple differential inequality due to I. Herbst, E. B. Davies and B. Simon, the author presents short and simple proofs of exponential integrability of Lipschitz functions and bounds on the diameter \(D\) of \(M\) provided \(\rho_0 > 0\). It is shown in particular that if the Ricci curvature of \(M\) is bounded below, then \(D \leq c/ \min (1, \rho_0)\), where \(C\) only depends on the dimension of \(M\) and the bound on the Ricci curvature. The bound is sharp and improves upon a recent result of \textit{L. Saloff-Coste} [Colloq. Math. 67, No. 1, 109-121 (1994; Zbl 0816.53027)].
Mathematics Subject Classification ID: 60H25
Mathematics Subject Classification ID: 58J65
zbMATH DE Number: 836411
zbMATH Keywords: logarithmic Sobolev inequality
zbMATH Keywords: exponential integrability
zbMATH Keywords: diameter
zbMATH Keywords: Ricci curvature
author: Michel Ledoux
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250518769
OpenAlex ID: W1506465796
DOI: 10.1215/KJM/1250518769
description: scientific article; zbMATH DE number 836411