Cheng-Yau Type Local Gradient Estimate for Harmonic Functions on RCD*(K,N) Metric Measure Spaces

Abstract

We present a geometric argument to prove a Cheng-Yau type local gradient estimate for positive harmonic functions on RCD*(K,N) metric measure spaces. The proof is an extension of Lihe Wang's geometric De Giorgi argument in [8] from Euclidean space to RCD*(K,N) space and uses a local uniform sobolev inequality and a local Bochner inequality established by Hua, Kell, and Xia in [6].