Uniform rectifiability from Carleson measure estimates and {\(\epsilon\)}-approximability of bounded harmonic functions (Q1645024)

Let \(\Omega \subset {\mathbb R}^{n+1}\), \(n\geq 1\), be a corkwise domain with \(n\)-Ahlfors-David-regular boundary. The authors show that the following conditions are equivalent. (a) \(\partial\Omega\) is uniformly \(n\)-rectifiable. (b) Every bounded harmonic function \(u\) on \(\Omega\) is \(\varepsilon\)-approximable for all \(\varepsilon >0\), i.e., there exists \(\phi \in W^{1,1}(\Omega)\) and \(C>0\) such that \(\|u-\phi\|_{L^{\infty}(\Omega)} <\varepsilon\) and for all \(x\in \partial\Omega\) and all \(r>0\) \[ \frac{1}{r^n}\int_{B(x,r)}|\nabla\phi(y)|\,dy \leq C. \] (c) There is \(C>0\) such that if \(u\) is a bounded harmonic function on \(\Omega\) and \(B\) is a ball centered on \(\partial\Omega\) with radius \(r\), then the Carleson measure estimate \[ \int_{B}|\nabla u(x)|^{2}\operatorname{dist}(x, \partial\Omega)\,dx \leq Cr^{n}\|u\|^{2}_{L^{\infty}(\Omega)} \] is valid.