Measure Estimates, Harnack Inequalities and Ricci Lower Bound

Yu Wang,

Xiangwen Zhang

 On the Riemannian metric-measure space, we establish an Alexandrov–Bakelman–Pucci type estimate connecting the Bakry–´Emery Ricci curvature lower bound, the modified Laplacian and the measure of certain special sets. We apply this estimate to prove the Harnack inequalities for the modified Laplacian operator (and fully non-linear operators, see the Appendix). These inequalities seem not available in the literature and our proof, based solely on the ABP estimate, does not use standard techniques.
Słowa kluczowe: ABP estimate, Krylov-Safonov Harnack inequality, metric- measure space, Bakry-Emery Ricci curvature, weighted Laplacian operator, Pucci operator

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