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

TL;DR

This paper establishes equivalences between various regularity and heat kernel conditions for harmonic functions on metric measure spaces, and applies these results to fractal-like systems, extending classical estimates.

Contribution

It introduces a H"older regularity condition and proves its equivalence with heat kernel estimates under certain geometric conditions, advancing analysis on fractal and metric spaces.

Findings

Proves equivalence of H"older regularity and heat kernel conditions.

Establishes generalized reverse H"older inequality on Sierpiński carpet cable system.

Derives heat kernel gradient estimates on fractal-like 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.