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

Title: Uniform rectifiability, Carleson measure estimates, and approximation of harmonic functions.

Author(s): Steve Hofmann, José María Martell and Svitlana Mayboroda.

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

Date of publication: 2016.

Abstract: Let EÌRn+1, n³2, be a uniformly rectifiable set of dimension n. Then bounded harmonic functions in \Omega:=R^{n+1}\backslash E satisfy Carleson measure estimates and are \epsilon-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.