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arXiv · 1408.1447

Uniform Rectifiability, Carleson measure estimates, and approximation of harmonic functions

Steve Hofmann·Jose Maria Martell·Svitlana Mayboroda
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Abstract

Let $E\subset \mathbb{R}^{n+1}$, $n\ge 2$, be a uniformly rectifiable set of dimension $n$. Then bounded harmonic functions in $Ω:= \mathbb{R}^{n+1}\setminus E$ 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.

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Steve Hofmann, Jose Maria Martell, Svitlana Mayboroda. 2014-08-07. Uniform Rectifiability, Carleson measure estimates, and approximation of harmonic functions. https://doi.org/10.1215/00127094-3477128

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