Uniform rectifiability from Carleson measure estimates and $\varepsilon$-approximability of bounded harmonic functions

TL;DR

This paper establishes that the boundary of certain domains is uniformly rectifiable if bounded harmonic functions meet specific approximation or Carleson measure criteria, confirming a conjecture and providing new geometric characterizations.

Contribution

It proves that Carleson measure estimates and $ ext{varepsilon}$-approximability imply uniform rectifiability, confirming a conjecture and introducing new criteria involving harmonic measure and $S<N$ estimates.

Findings

Uniform rectifiability follows from Carleson measure estimates.

$ ext{varepsilon}$-approximability of harmonic functions implies rectifiability.

New criteria for rectifiability using harmonic measure and $S<N$ estimates.

Abstract

Let $\Omega\subset\mathbb R^{n+1}$, $n\geq1$, be a corkscrew domain with Ahlfors-David regular boundary. In this paper we prove that $\partial\Omega$ is uniformly $n$-rectifiable if every bounded harmonic function on $\Omega$ is $\varepsilon$-approximable or if every bounded harmonic function on $\Omega$ satisfies a suitable square-function Carleson measure estimate. In particular, this applies to the case when $\Omega=\mathbb R^{n+1}\setminus E$ and $E$ is Ahlfors-David regular. Our results solve a conjecture posed by Hofmann, Martell, and Mayboroda in a recent work where they proved the converse statements. Here we also obtain two additional criteria for uniform rectifiability. One is given in terms of the so-called "$S<N$" estimates, and another in terms of a suitable corona decomposition involving harmonic measure.