Measure Estimates, Harnack Inequalities and Ricci Lower Bound

Details Measure Estimates, Harnack Inequalities and Ricci Lower Bound Measure Estimates, Harnack Inequalities and Ricci Lower Bound

2011-02-28

arxiv.org/abs/1102.5567v2

Mathematics

Analysis of PDEs

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

Scientific paper

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.

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