On harmonic measure and rectifiability in uniform domains (Q2419729)

Let \(\omega \) denote harmonic measure for a domain \(\Omega \) in \(\mathbb{R}^{d+1}\), where \(d\geq 1\). There is an extensive literature investigating conditions on \(\partial \Omega \) which ensure that the \(d\)-dimensional Hausdorff measure \(\mathcal{H}^{d}|_{\partial \Omega }\) satisfies \(\mathcal{H}^{d}|_{\partial \Omega }\ll \omega \), or \(\omega \ll \mathcal{H}^{d}|_{\partial \Omega }\), or both. The present paper shows that, if \(\Omega \) is a uniform domain with lower \(d\)-Ahlfors-David regular and \(d\)-rectifiable boundary, and if \(\mathcal{H}^{d}|_{\partial \Omega }\) is locally finite, then \(\mathcal{H}^{d}|_{\partial \Omega }\ll \omega \). The author notes that a closely related result was independently obtained by \textit{M. Akman} et al. [Trans. Am. Math. Soc. 369, No. 8, 5711--5745 (2017; Zbl 1373.31003)].