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Hölder regularity of harmonic functions on metric measure spaces

Jin Gao, Meng Yang

TL;DR

This paper develops a Hölder regularity theory for harmonic functions on unbounded metric measure Dirichlet spaces with two-scale growth and proves an equivalence among $HR$, $wBE$, $HHK$, $HHK_{exp}$, and $NLE$ under upper heat-kernel bounds $UHK(\Psi)$ and slow volume regularity $V(x,r)\asymp \Phi(r)$, assuming strongly recurrent walk dimensions $\alpha_i<\beta_i$. The approach leverages Poisson-type Hölder estimates, Morrey–Sobolev inequalities, and Kigami’s resistance-form framework to connect harmonic regularity with heat-kernel regularity and lower bounds, avoiding gradient-operator assumptions. It yields gradient estimates for the heat kernel from two-sided heat-kernel bounds, confirms generalized reverse Hölder inequalities on the Sierpiński carpet cable system, and extends Li–Yau type Hölder/Lipschitz estimates to strongly recurrent fractal-like spaces. The results are quasi-isometry stable and illuminate heat-kernel behavior on fractal-like metric measure spaces, with applications to fractal cables and blowups of fractals.

Abstract

We introduce a Hölder 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 Hölder regularity condition, the weak Bakry-Émery non-negative curvature condition, Hölder 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 Hölder inequality on the Sierpiński 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 Hölder (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.

Hölder regularity of harmonic functions on metric measure spaces

TL;DR

This paper develops a Hölder regularity theory for harmonic functions on unbounded metric measure Dirichlet spaces with two-scale growth and proves an equivalence among , , , , and under upper heat-kernel bounds and slow volume regularity , assuming strongly recurrent walk dimensions . The approach leverages Poisson-type Hölder estimates, Morrey–Sobolev inequalities, and Kigami’s resistance-form framework to connect harmonic regularity with heat-kernel regularity and lower bounds, avoiding gradient-operator assumptions. It yields gradient estimates for the heat kernel from two-sided heat-kernel bounds, confirms generalized reverse Hölder inequalities on the Sierpiński carpet cable system, and extends Li–Yau type Hölder/Lipschitz estimates to strongly recurrent fractal-like spaces. The results are quasi-isometry stable and illuminate heat-kernel behavior on fractal-like metric measure spaces, with applications to fractal cables and blowups of fractals.

Abstract

We introduce a Hölder 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 Hölder regularity condition, the weak Bakry-Émery non-negative curvature condition, Hölder 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 Hölder inequality on the Sierpiński 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 Hölder (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.
Paper Structure (9 sections, 22 theorems, 185 equations, 4 figures)

This paper contains 9 sections, 22 theorems, 185 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Statement of the main results
  3. Heat kernel estimates
  4. Poisson equation on unbounded MMD spaces
  5. Proof of "HR implies wBE"
  6. Proof of "HR implies HHKexp"
  7. Proof of "wBE implies HHK implies NLE"
  8. Proof of "NLE implies HR"
  9. Some examples related to the Sierpiński carpet and the Vicsek set

Key Result

Theorem 2.1

Let $(X,d,m,\mathcal{E},\mathcal{F})$ be an unbounded MMD space satisfying eq_VPhi and eq_UHK, where $\Phi$ and $\Psi$ are given by (eq_Phi) and (eq_Psi), respectively, with parameters $\alpha_i,\beta_i$, $i=1,2$, satisfying the strongly recurrent condition (eq_recurrent). Then the followings are eq

Figures (4)

  • Figure 1: The Sierpiński gasket
  • Figure 2: The Sierpiński carpet
  • Figure 3: The correspondence between Hölder-type and gradient-type conditions
  • Figure 4: $V^{(0)}$, $V^{(1)}$ and $V^{(2)}$

Theorems & Definitions (51)

  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Corollary 2.6
  • Remark 2.7
  • Corollary 2.8
  • Remark 2.9
  • Corollary 2.10
  • ...and 41 more