Uniform Rectifiability, Carleson measure estimates, and approximation of harmonic functions

TL;DR

This paper proves that harmonic functions in domains with uniformly rectifiable boundaries satisfy Carleson measure estimates and can be approximated, extending classical results to more general geometric settings.

Contribution

It establishes that harmonic functions in domains with uniformly rectifiable boundaries satisfy key estimates and approximation properties, generalizing classical theorems.

Findings

Harmonic functions satisfy Carleson measure estimates in these domains.

Harmonic functions are $ ext{ extquoteright} ext{ extquoteright} ext{ extquoteright}$-approximable.

Results relate harmonic measure to surface measure in these settings.

Abstract

Let $E\subset \mathbb{R}^{n+1}$, $n\ge 2$, be a uniformly rectifiable set of dimension $n$. Then bounded harmonic functions in $\Omega:= \mathbb{R}^{n+1}\setminus E$ satisfy Carleson measure estimates, and are "$\varepsilon$-approximable". Our results may be viewed as generalized versions of the classical F. and M. Riesz theorem, since the estimates that we prove are equivalent, in more topologically friendly settings, to quantitative mutual absolute continuity of harmonic measure, and surface measure.