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The Hölder regularity of harmonic function on bounded and unbounded p.c.f self-similar sets

Jin Gao, Yijun Song

Abstract

In this paper, we prove a generalized reverse Hölder inequality of harmonic functions on cable systems induced by post-critically finite (p.c.f.) self-similar sets. Furthermore, we also establish the Hölder regularity of harmonic functions on both bounded and unbounded p.c.f. self-similar sets, which does not involve heat kernel estimates and resistance estimates.

The Hölder regularity of harmonic function on bounded and unbounded p.c.f self-similar sets

Abstract

In this paper, we prove a generalized reverse Hölder inequality of harmonic functions on cable systems induced by post-critically finite (p.c.f.) self-similar sets. Furthermore, we also establish the Hölder regularity of harmonic functions on both bounded and unbounded p.c.f. self-similar sets, which does not involve heat kernel estimates and resistance estimates.
Paper Structure (5 sections, 9 theorems, 95 equations, 5 figures)

This paper contains 5 sections, 9 theorems, 95 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Definitions and the Main Result
  3. The oscillation inequality of harmonic function
  4. Proof of Theorem \ref{['thm_GRH']} and Theorem \ref{['corol_GRH']}
  5. Example

Key Result

Theorem 2.1

Let $X$ be a cable system induced by p.c.f. self-similar set $K$. We have that generalized reverse Hölder inequality eq_GRH holds on $X$.

Figures (5)

  • Figure 1: The Sierpiński Gasket
  • Figure 2: The Vicsek set
  • Figure 3: harmonic function $u$ on $V_0$ for the Sierpiński gasket
  • Figure 4: harmonic function $u$ on $V_0$ for the Vicsek set
  • Figure 5: A eyebolted Vicsek cross

Theorems & Definitions (23)

  • Theorem 2.1
  • Theorem 2.2
  • Corollary 2.3
  • Remark 2.4
  • Remark 2.5
  • Lemma 3.1
  • Remark 3.2
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • ...and 13 more