Uniform rectifiability, Carleson measure estimates, and approximation of harmonic functions (Q335022)

The authors of the present paper generalize versions of the classical Riesz theorem [\textit{F. Riesz} and \textit{M. Riesz}, Quatrième congrès des math. scand. 1916, 27--44 (1916; JFM 47.0295.03)], since the derived estimates are equivalent in more topologically friendly domains to quantitative mutual absolute continuity of harmonic measure and surface measure. The main results are {\parindent=0.6cm\begin{itemize}\item[--] For a simply connected domain in the complex plane, with a rectifiable boundary, harmonic measure is absolutely continuous with respect to arclength measure. \item[--] Let \(E\subset \mathbb{R}^{n+1}\) be a uniformly rectifiable set of codimension \(1\) and suppose \(u\) is harmonic and bounded in \( \mathbb{R}^{n+1}E\), then \(u\) satisfies Carleson measure estimates and is \(\epsilon\)-approximable. NEWLINENEWLINE\end{itemize}} These new results are scale-invariant, higher-dimensional versions of the Riesz theorem and hold even in the absence of connectivity. A general mechanism that permits eliminating the hypothesis of connectivity and still getting Carleson measure bounds and \(\epsilon\)-approximability is actually introduced in this paper.