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Bounding Radial Variation of positive harmonic Functions on Lipschitz Domains

Jakob Fromherz, Paul F. X. Müller, Katharina Riegler

TL;DR

The paper addresses the problem of bounding the radial variation of positive harmonic functions on Lipschitz domains by adapting Mozolyako–Havin's approach to Lipschitz geometry through a near half-space framework. It develops a kernel toolkit (including $k_y$, $c_y$, $b_y$, and $\omega_\Delta$) and a dual measure $\nu_\varepsilon$ to control vertical variation $V(x)$ and identify Bourgain points within surface balls, culminating in a main near-half-space result that guarantees points with bounded variation relative to a Poisson-type kernel. The work also proves a Hausdorff-dimension lower bound $\dim_H(\mathcal{V}) \ge (d-1)\frac{1+\eta}{2+\eta}$, where $\eta>0$ depends on the boundary, and extends these results to general Lipschitz domains via local flattening and Dahlberg-type harmonic-measure theory. Overall, the paper advances higher-dimensional Bourgain-point theory on Lipschitz domains by refining multi-scale kernel methods and duality arguments, enabling sharper dimension estimates and a path to full Lipschitz-domain generalization.

Abstract

We provide radial variational estimates for positive harmonic functions on Lipschitz domains in higher dimensions. The intention of this paper is to document an updated and refined version of arXiv:2003.07176 which modifies the proof of Mozolyako and Havin for Lipschitz domains.

Bounding Radial Variation of positive harmonic Functions on Lipschitz Domains

TL;DR

The paper addresses the problem of bounding the radial variation of positive harmonic functions on Lipschitz domains by adapting Mozolyako–Havin's approach to Lipschitz geometry through a near half-space framework. It develops a kernel toolkit (including , , , and ) and a dual measure to control vertical variation and identify Bourgain points within surface balls, culminating in a main near-half-space result that guarantees points with bounded variation relative to a Poisson-type kernel. The work also proves a Hausdorff-dimension lower bound , where depends on the boundary, and extends these results to general Lipschitz domains via local flattening and Dahlberg-type harmonic-measure theory. Overall, the paper advances higher-dimensional Bourgain-point theory on Lipschitz domains by refining multi-scale kernel methods and duality arguments, enabling sharper dimension estimates and a path to full Lipschitz-domain generalization.

Abstract

We provide radial variational estimates for positive harmonic functions on Lipschitz domains in higher dimensions. The intention of this paper is to document an updated and refined version of arXiv:2003.07176 which modifies the proof of Mozolyako and Havin for Lipschitz domains.

Paper Structure

This paper contains 25 sections, 41 theorems, 281 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Harnack Inequality and Harnack chains
  5. The Dirichlet problem and Harmonic Measure
  6. Harmonic Measure and Hausdorff Measure: Dahlberg's Theorem
  7. The Martin kernel for epigraphs of Lipschitz domains
  8. Radial Variation on Near Half Spaces
  9. Kernels
  10. Continuity of $b_y$
  11. Variation and the set $\mathcal{V}(S)$
  12. Main Theorem For Near Half Spaces
  13. Notation for Partitions
  14. Construction of the kernel $\omega_{\Delta}$
  15. Weak $\lambda$-regularity
  16. ...and 10 more sections

Key Result

Theorem 2.2

Let $x_0\in\mathbb{R}^d$, $R>0$ and $u:B(x_0,R)\longrightarrow\mathbb{R}$ be non-negative harmonic function. Then for every $x\in\mathbb{R}^d$ with $\|x-x_0\|=r<R$ we have the double inequality

Figures (5)

  • Figure 1: A Harnack chain connecting $x_{y_2}$ and $x_{y_1}$
  • Figure 2: Overview of statement in Main Theorem \ref{['thm:main']}.
  • Figure 3: $\Lambda:= \frac{c}{\omega^{z_0}\left(B(x,y)\cap S\right)}$
  • Figure 4: Identifying the Near Half Space $\mathcal{O}$ and the special Lipschitz domain $U$.
  • Figure 5: Schematic representation of sets in above proof.

Theorems & Definitions (100)

  • Definition 2.1: Near Half Space
  • Remark 1
  • Theorem 2.2: Harnack inequality Gilbarg2001
  • Corollary 2.2.1
  • proof
  • Lemma 2.3: MuellerRiegler2020
  • proof
  • Lemma 2.4
  • proof
  • Definition 2.5
  • ...and 90 more