Measure Estimates, Harnack Inequalities and Ricci Lower Bound
arXiv:1102.5567
Abstract
On a Riemannian metric-measure space, we establish an Alexandrov-Bakelman-Pucci type measure estimate connecting Bakry-Émery 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.
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