H\"older regularity of harmonic functions on metric measure spaces

Abstract

We introduce a H\"older regularity condition for harmonic functions on metric measure spaces and prove that, under a slow volume regular condition and an upper heat kernel estimate, the H\"older regularity condition, the weak Bakry-\'Emery non-negative curvature condition, H\"older continuity of the heat kernel (with or without exponential terms), and the near-diagonal lower bound for the heat kernel are equivalent. As applications, first, we establish the validity of the so-called generalized reverse H\"older inequality on the Sierpi\'nski carpet cable system, resolving an open problem left by Devyver, Russ, Yang (Int. Math. Res. Not. IMRN (2023), no. 18, 15537-15583). Second, we prove that two-sided heat kernel estimates alone imply gradient estimates for the heat kernel on strongly recurrent fractal-like cable systems, improving the main results of the aforementioned paper. Third, we obtain H\"older (Lipschitz) estimates for the heat kernel on strongly recurrent metric measure spaces, extending the classical Li-Yau gradient estimate for the heat kernel on Riemannian manifolds.