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On the dimension drop for harmonic measure on uniformly non-flat Ahlfors-David regular boundaries

Aritro Pathak

TL;DR

This work studies the dimension drop of harmonic measure for domains in $\mathbb{R}^n$ with boundary that is $s$-AD regular and satisfies a uniform non-flatness (bilateral) condition, with $n-1-\delta_0\le s\le n-1$. The authors present an elementary potential-theoretic approach, avoiding Riesz transforms and compactness arguments, to prove that $\dim \omega_{\Omega} < \kappa s$ for some $\kappa\in(0,1)$, providing explicit bounds on the small parameter $\delta_0 = δ_0(β,C_1)$. The strategy hinges on a localized Green function analysis, a corkscrew-based pole construction, a dyadic cube decomposition on the boundary, and a change-of-pole mechanism that yields scale-by-scale density growth/decay and hence a dimensional restriction. The results extend prior work on dimension drop for non-flat boundaries and open avenues for applications to uniform rectifiability and absolute continuity of harmonic measure on rough domains.

Abstract

We extend earlier results of Azzam on the dimension drop of the harmonic measure for a domain $Ω\subset \R^{n}$ with $n\geq 3$, with dimensional Ahlfors regular boundary $\partialΩ$ of dimension $s$ with $n-1-δ_0 \leq s\leq n-1$, that is uniformly non flat. Here $δ_0$ is a small positive constant dependent on the parameters of the problem. Our novel construction relies on elementary geometric and potential theoretic considerations. We avoid the use of Riesz transforms and compactness arguments, and also give quantitative bounds on the $δ_0$ parameter.

On the dimension drop for harmonic measure on uniformly non-flat Ahlfors-David regular boundaries

TL;DR

This work studies the dimension drop of harmonic measure for domains in with boundary that is -AD regular and satisfies a uniform non-flatness (bilateral) condition, with . The authors present an elementary potential-theoretic approach, avoiding Riesz transforms and compactness arguments, to prove that for some , providing explicit bounds on the small parameter . The strategy hinges on a localized Green function analysis, a corkscrew-based pole construction, a dyadic cube decomposition on the boundary, and a change-of-pole mechanism that yields scale-by-scale density growth/decay and hence a dimensional restriction. The results extend prior work on dimension drop for non-flat boundaries and open avenues for applications to uniform rectifiability and absolute continuity of harmonic measure on rough domains.

Abstract

We extend earlier results of Azzam on the dimension drop of the harmonic measure for a domain with , with dimensional Ahlfors regular boundary of dimension with , that is uniformly non flat. Here is a small positive constant dependent on the parameters of the problem. Our novel construction relies on elementary geometric and potential theoretic considerations. We avoid the use of Riesz transforms and compactness arguments, and also give quantitative bounds on the parameter.
Paper Structure (9 sections, 13 theorems, 151 equations)

This paper contains 9 sections, 13 theorems, 151 equations.

Table of Contents

  1. Introduction.
  2. Basic estimates and overview of the proof.
  3. Preliminaries
  4. Notation
  5. Proof of the theorem.
  6. Change of poles argument.
  7. Dimension drop of the harmonic measure
  8. Future directions
  9. Acknowledgements

Key Result

Theorem 1

Given any integer $n \geq 3$, $C_{1} > 1$, and $0<\beta <1$, there is a constant $\kappa \in (0,1)$, so that the following holds. Define, $N_\beta =\lceil \frac{1}{\beta} \rceil$ and $\beta_1:= \frac{1}{4(N_{\beta}+1)}$. Suppose Suppose $\Omega \subseteq \mathbb{R}^{n}$ is a connected domain. Further, for any point $x\in \mathbb{R}^n$ and any $r< \text{diam}(\Omega)$, we have, where the infimum

Theorems & Definitions (25)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Definition 1
  • Definition 2
  • ...and 15 more