The H\"older regularity of harmonic function on bounded and unbounded p.c.f self-similar sets

TL;DR

This paper proves the H"older regularity of harmonic functions on p.c.f. self-similar sets using a generalized reverse H"older inequality, without relying on heat kernel or resistance estimates.

Contribution

It introduces a new approach to establish H"older regularity of harmonic functions on p.c.f. sets without traditional heat kernel or resistance estimates.

Findings

Harmonic functions satisfy a generalized reverse H"older inequality.

H"older regularity is established on both bounded and unbounded p.c.f. sets.

Method avoids reliance on heat kernel and resistance estimates.

Abstract

In this paper, we prove a generalized reverse H\"older inequality of harmonic functions on cable systems induced by post-critically finite (p.c.f.) self-similar sets. Furthermore, we also establish the H\"older 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.