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

TL;DR

This paper proves a dimension drop phenomenon for harmonic measure on certain non-flat boundaries, extending previous results and providing explicit bounds without complex analytical tools.

Contribution

It introduces a new geometric and potential theoretic approach to demonstrate the dimension drop for harmonic measure on uniformly non-flat Ahlfors-David regular boundaries, avoiding Riesz transforms.

Findings

Dimension drop occurs for harmonic measure on non-flat boundaries.

Explicit bounds on the parameter  are provided.

Method relies on elementary geometric and potential theoretic considerations.

Abstract

We extend earlier results of Azzam on the dimension drop of the harmonic measure for a domain $\Omega\subset \R^{n}$ with $n\geq 3$, with dimensional Ahlfors regular boundary $\partial\Omega$ of dimension $s$ with $n-1-\delta_0 \leq s\leq n-1$, that is uniformly non flat. Here $\delta_0$ 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 $\delta_0$ parameter.