## Duke Mathematical Journal

### Uniform rectifiability, Carleson measure estimates, and approximation of harmonic functions

#### Abstract

Let $E\subset\mathbb{R}^{n+1}$, $n\ge2$, be a uniformly rectifiable set of dimension $n$. Then bounded harmonic functions in $\Omega:=\mathbb{R}^{n+1}\setminus E$ satisfy Carleson measure estimates and are $\varepsilon$-approximable. Our results may be viewed as generalized versions of the classical F. and M. Riesz theorem, since the estimates that we prove are equivalent, in more topologically friendly settings, to quantitative mutual absolute continuity of harmonic measure and surface measure.

#### Article information

Source
Duke Math. J. Volume 165, Number 12 (2016), 2331-2389.

Dates
Received: 6 August 2014
Revised: 2 July 2015
First available in Project Euclid: 6 September 2016

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1473186402

Digital Object Identifier
doi:10.1215/00127094-3477128

Mathematical Reviews number (MathSciNet)
MR3544283

Zentralblatt MATH identifier
06646643

Subjects
Primary: 28A75: Length, area, volume, other geometric measure theory [See also 26B15, 49Q15]
Secondary: 28A78: Hausdorff and packing measures 31B05: Harmonic, subharmonic, superharmonic functions 35J25: Boundary value problems for second-order elliptic equations 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B25: Maximal functions, Littlewood-Paley theory 42B37: Harmonic analysis and PDE [See also 35-XX]

Keywords
Carleson measures square functions nontangential maximal functions $\varepsilon$-approximability harmonic measure Poisson kernel uniform rectifiability harmonic functions

#### Citation

Hofmann, Steve; Martell, José María; Mayboroda, Svitlana. Uniform rectifiability, Carleson measure estimates, and approximation of harmonic functions. Duke Math. J. 165 (2016), no. 12, 2331--2389. doi:10.1215/00127094-3477128. https://projecteuclid.org/euclid.dmj/1473186402

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