Uniform rectifiability from Carleson measure estimates and ε-approximability of bounded harmonic functions

Abstract

Let $\Omega \subset {\mathbb{R}}^{n+1}$, $n\ge 1$, be a corkscrew domain with Ahlfors–David regular boundary. In this article we prove that $\partial \Omega$ is uniformly $n$-rectifiable if every bounded harmonic function on $\Omega$ is $\epsilon$-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 establish a conjecture posed by Hofmann, Martell, and Mayboroda, in which they proved the converse statements. Here we also obtain two additional criteria for uniform rectifiability, one in terms of the so-called $S<N$ estimates and another in terms of a suitable corona decomposition involving harmonic measure.