## Duke Mathematical Journal

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

#### Abstract

Let $\Omega\subset{\mathbb{R}}^{n+1}$, $n\geq1$, 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 $\varepsilon$-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 $\Omega={\mathbb{R}}^{n+1}\setminusE$ 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 $S\lt N$ estimates and another in terms of a suitable corona decomposition involving harmonic measure.

#### Article information

Source
Duke Math. J., Volume 167, Number 8 (2018), 1473-1524.

Dates
Revised: 27 September 2017
First available in Project Euclid: 3 May 2018

https://projecteuclid.org/euclid.dmj/1525313238

Digital Object Identifier
doi:10.1215/00127094-2017-0057

Mathematical Reviews number (MathSciNet)
MR3807315

Zentralblatt MATH identifier
06896951

#### Citation

Garnett, John; Mourgoglou, Mihalis; Tolsa, Xavier. Uniform rectifiability from Carleson measure estimates and $\mathbf{\varepsilon}$ -approximability of bounded harmonic functions. Duke Math. J. 167 (2018), no. 8, 1473--1524. doi:10.1215/00127094-2017-0057. https://projecteuclid.org/euclid.dmj/1525313238

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