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1 June 2018 Uniform rectifiability from Carleson measure estimates and ε-approximability of bounded harmonic functions
John Garnett, Mihalis Mourgoglou, Xavier Tolsa
Duke Math. J. 167(8): 1473-1524 (1 June 2018). DOI: 10.1215/00127094-2017-0057

Abstract

Let ΩRn+1, n1, be a corkscrew domain with Ahlfors–David regular boundary. In this article we prove that Ω is uniformly n-rectifiable if every bounded harmonic function on Ω is ε-approximable or if every bounded harmonic function on Ω satisfies a suitable square-function Carleson measure estimate. In particular, this applies to the case when Ω=Rn+1E 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.

Citation

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John Garnett. Mihalis Mourgoglou. Xavier Tolsa. "Uniform rectifiability from Carleson measure estimates and ε-approximability of bounded harmonic functions." Duke Math. J. 167 (8) 1473 - 1524, 1 June 2018. https://doi.org/10.1215/00127094-2017-0057

Information

Received: 4 December 2016; Revised: 27 September 2017; Published: 1 June 2018
First available in Project Euclid: 3 May 2018

zbMATH: 06896951
MathSciNet: MR3807315
Digital Object Identifier: 10.1215/00127094-2017-0057

Subjects:
Primary: 28A75
Secondary: 26B15 , 28A78 , 31A15 , 31B05 , 35J25 , 49Q15

Keywords: Carleson measures , uniform rectifiability , ε-approximation

Rights: Copyright © 2018 Duke University Press

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Vol.167 • No. 8 • 1 June 2018
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