Harmonic functions on metric measure spaces
Bobo Hua, Martin Kell,and Chao Xia
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Submission date: 18. Feb. 2014
MSC-Numbers: 30L99, 31B05
Keywords and phrases: harmonic function, metric measure space
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In this paper, we study harmonic functions on metric measure spaces with Riemannian Ricci curvature bounded from below, which were introduced by Ambrosio-Gigli-Savare. We prove a Cheng-Yau type local gradient estimate for harmonic functions on these spaces. Furthermore, we derive various optimal dimension estimates for spaces of polynomial growth harmonic functions on metric measure spaces with nonnegative Riemannian Ricci curvature.