A generalization of Poincaré and log-Sobolev inequalities
type: Item
instance of: scholarly article
title: A generalization of Poincaré and log-Sobolev inequalities
zbMATH Open document ID: 1068.47051
author: Feng-Yu Wang
published in: Potential Analysis
publication date: 2005-05-13T00:00:00Z
review text: Conditions for the validity of the relation \[ \sup_{p\in[1,2)} \{[\mu(f^2)-\mu(| f| ^p)^{2/p}]/\varphi(p)\}\leq\varepsilon(f,f), \quad f\in D(L), \] are provided, where \(\varepsilon(f,f)=-\mu(fLf)\), \(L\) generates in \(L^2(\mu)\) a Markov semigroup with invariant probability measure \(\mu\), and \(\varphi\in C[1,2]\) is a decreasing function, \(\varphi(p)>0\) for \(p<2\). The inequality studied by \textit{R. Latala} and \textit{K. Oleszkiewicz} in [Lect. Notes Math. 1745, 147--168 (2000; Zbl 0986.60017)] is recovered when \(\varphi(p)=C(p-2)^{\alpha}\), \(C>0\), and \(\alpha\in [1,2]\). Connections with other relevant inequalities having an ``additivity'' property are discussed. Thereby, some results on the exponential convergence of Markov semigroups are generalized.
Mathematics Subject Classification ID: 47D07
zbMATH DE Number: 2168217
zbMATH Keywords: functional inequality
zbMATH Keywords: Markov semigroup
zbMATH Keywords: concentration
reviewed by: Andrey V. Bulinski
MaRDI profile type: Publication
full work available at URL: s11118-003-4006-0
OpenAlex ID: W2049330328
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DOI: 10.1007/S11118-003-4006-0
description: scientific article; zbMATH DE number 2168217