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

Steve Hofmann, José María Martell, Svitlana Mayboroda

Type: Article

Publication Date: 2016-06-23

Citations: 55

DOI: https://doi.org/10.1215/00127094-3477128

Abstract

Let $E\subset \mathbb{R}^{n+1}$, $n\ge 2$, 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.

Locations

Duke Mathematical Journal - View
arXiv (Cornell University) - View - PDF
DIGITAL.CSIC (Spanish National Research Council (CSIC)) - View - PDF
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