Weak curvature conditions and functional inequalities
type: Item
instance of: scholarly article
title: Weak curvature conditions and functional inequalities
zbMATH Open document ID: 1119.53028
author: John Lott
author: Cédric Villani
published in: Journal of Functional Analysis
publication date: 2007-05-10T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0506481
review text: In the paper the authors give sufficient conditions for a measured length space \((X, d,\nu)\) to admit local and global Poincare inequalities. lt is a continuation of the paper of authors entitled Ricci curvature for metric-measure spaces via optimal transport which will appear in Ann. Math.. Following on the work of Cheeger and Colding they introduce a certain condition DM on a measured length space. Roughly speaking the condition DM says that there is a dynamical democratic transference plan so that a given point is not hit too often by the geodesics. Then they show that this condition together with a doubling condition on \(\nu\), implies a scale invariant local Poincaré inequality. They show that if \((X, d, \nu)\) has nonnegative \(N\)-Ricci curvature and has unique minimizing geodesics between almost all pairs of points then it satisfies DM, with constant \(2^N\). The condition DM is preserved by measured Gromov-Hausdorff limits. Based on it they prove a Sobolev inequality for measured length spaces with \(N\)-Ricci curvature bounded below by \(K > 0\). Finally the authors derive a sharp global Poincaré inequality.
reviewed by: Andrzej Szczepański
Mathematics Subject Classification ID: 53C23
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 31C15
Mathematics Subject Classification ID: 49Q15
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 58C35
zbMATH DE Number: 5151645
zbMATH Keywords: Poincaré inequalities
zbMATH Keywords: Ricci curvature
zbMATH Keywords: metric-measure spaces
zbMATH Keywords: Sobolev inequality
zbMATH Keywords: Gromov-Hausdorff limits
MaRDI profile type: Publication
OpenAlex ID: W2164089941
arXiv ID: math/0506481
Wikidata QID: Q125775773
cites work: A course in metric geometry
cites work: Differentiability of Lipschitz functions on metric measure spaces
cites work: Lower bounds on Ricci curvature and the almost rigidity of warped products
cites work: On the structure of spaces with Ricci curvature bounded below. III
cites work: A Riemannian interpolation inequality à la Borell, Brascamp and Lieb
cites work: Lectures on analysis on metric spaces
cites work: Modulus and the Poincaré inequality on metric measure spaces
cites work: Q4661855
cites work: Q3274645
cites work: Ricci curvature for metric-measure spaces via optimal transport
cites work: On local Poincaré via transportation
cites work: On the geometry of metric measure spaces. I
cites work: On the geometry of metric measure spaces. II
cites work: Q4805362
cites work: Q3687381
DOI: 10.1016/J.JFA.2006.10.018
description: scientific article