Measure Estimates, Harnack Inequalities and Ricci Lower Bound
Analysis of PDEs
2011-04-12 v2 Differential Geometry
Abstract
On a Riemannian metric-measure space, we establish an Alexandrov-Bakelman-Pucci type measure estimate connecting Bakry-\'Emery Ricci curvature lower bound, modified Laplacian and the measure of certain special sets. We apply this estimate to prove Harnack inequalities for the modified Laplacian operator and fully non-linear operators. These inequalities seem not available in the literature; And our proof, solely based on the ABP estimate, does not involve any Sobolev inequalities nor gradient estimate. We also propose a question regarding the characterization of Ricci lower bound by the Harnack inequality.
Keywords
Cite
@article{arxiv.1102.5567,
title = {Measure Estimates, Harnack Inequalities and Ricci Lower Bound},
author = {Yu Wang and Xiangwen Zhang},
journal= {arXiv preprint arXiv:1102.5567},
year = {2011}
}
Comments
37 pages. We fix an error in the proof of Lemma 9.3 in previous version which was pointed out to us by Prof. Diego Maldonado