Duke Mathematical Journal

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

John Garnett, Mihalis Mourgoglou, and Xavier Tolsa

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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.

Article information

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

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

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Zentralblatt MATH identifier

Primary: 28A75: Length, area, volume, other geometric measure theory [See also 26B15, 49Q15]
Secondary: 31A15: Potentials and capacity, harmonic measure, extremal length [See also 30C85] 26B15: Integration: length, area, volume [See also 28A75, 51M25] 49Q15: Geometric measure and integration theory, integral and normal currents [See also 28A75, 32C30, 58A25, 58C35] 28A78: Hausdorff and packing measures 31B05: Harmonic, subharmonic, superharmonic functions 35J25: Boundary value problems for second-order elliptic equations

uniform rectifiability ε-approximation Carleson measures


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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