[论文解读] The Hölder regularity of harmonic function on bounded and unbounded p.c.f self-similar sets

Jin Gao, Yijun Song|arXiv (Cornell University)|Feb 28, 2026

Nonlinear Partial Differential Equations被引用 0

一句话总结

论文证明了通过由 pcf 自相似集诱导的电缆系统上的调和函数梯度的广义逆 Hölder 不等式，并在有界与无界的 pcf 自相似集上建立 Hölder 正则性，且不依赖热核或电阻估计。

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.

研究动机与目标

• Motivate and analyze Hölder-type regularity for harmonic functions on post-critically finite (p.c.f.) self-similar sets.
• Develop intrinsic proofs that avoid heat kernel and resistance estimates for regularity results.
• Establish two main results: a generalized reverse Hölder inequality (GRH) on cable systems and Hölder regularity (HR) on both bounded and unbounded p.c.f. self-similar sets.

提出的方法

• Define Dirichlet forms and harmonic structures on p.c.f. self-similar sets and their cable systems.
• Prove an oscillation inequality (OSC) for harmonic functions using harmonic extension matrices and energy minimization.
• Derive GRH from harmonic extension matrices and energy estimates, leveraging combs of iterates of the harmonic extension operator.
• Prove HR by combining OSC with auxiliary estimates in the harmonic framework, without requiring heat kernel or resistance bounds.

实验结果

研究问题

• RQ1 Can condition GRH be established on cable systems induced by a class of p.c.f. self-similar sets?
• RQ2 Can the Hölder regularity condition HR be proved intrinsically on p.c.f. self-similar sets without heat kernel estimates?
• RQ3 How does the harmonic extension framework yield gradient control and oscillation bounds for harmonic functions on p.c.f. self-similar sets?
• RQ4 Do bounded and unbounded p.c.f. self-similar sets share HR under the presented harmonic-structure approach?

主要发现

• Generalized reverse Hölder inequality (GRH) holds on cable systems induced by p.c.f. self-similar sets.
• Hölder regularity (HR) holds on both bounded and unbounded p.c.f. self-similar sets using intrinsic harmonic structure.
• An oscillation inequality (OSC) is established for harmonic functions, underpinning HR.
• HR implies heat kernel Hölder continuity in unbounded p.c.f. self-similar sets under volume regular and upper heat kernel conditions.
• The proofs rely on harmonic extension matrices and energy-minimization arguments rather than heat kernel or resistance estimates.