Uniform rectifiability from Carleson measure estimates and $\epsilon$-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 paper 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 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.

Correction of a few typos and general reorganization of the arguments. Additional references

References in corpus (4)

• A new characterization of chord-arc domains
• Rectifiability of harmonic measure
• Uniform Rectifiability, Carleson measure estimates, and approximation of harmonic functions
• The weak-$A_{\infty }$ property of harmonic and $p$-harmonic measures implies uniform rectifiability

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• Interactions between quantitative rectifiability, singular integrals, and boundary value problems for harmonic functions
• Absolute continuity of degenerate elliptic measure
• Lipschitz decompositions of domains with bilaterally flat boundaries