Harmonic measure and approximation of uniformly rectifiable sets

TL;DR

This paper demonstrates that uniformly rectifiable sets in Euclidean space contain large portions of boundaries of chord-arc domains with controlled geometric properties, ensuring harmonic measure exhibits weak-$A_ abla_ ext{infty}$ behavior.

Contribution

It establishes a new connection between uniform rectifiability and the presence of chord-arc domain boundaries with harmonic measure properties.

Findings

Uniformly rectifiable sets contain big pieces of chord-arc domain boundaries.

Harmonic measure on these sets belongs to weak-$A_ abla_ ext{infty}$.

Domains satisfy a 2-sided corkscrew condition with controlled constants.

Abstract

Let $E\subset \mathbb{R}^{n+1}$, $n\ge 1$, be a uniformly rectifiable set of dimension $n$. We show $E$ that has big pieces of boundaries of a class of domains which satisfy a 2-sided corkscrew condition, and whose connected components are all chord-arc domains (with uniform control of the various constants). As a consequence, we deduce that $E$ has big pieces of sets for which harmonic measure belongs to weak-$A_\infty$