Rectifiability of harmonic measure (Q311134)

This paper concerns the relationship between harmonic measure \(\omega \) for a domain \(\Omega \subset \mathbb{R}^{n+1}\), \(n\geq 1\), and the geometry of \( \partial \Omega \). There are well-known results which say that \(\omega \) is absolutely continuous with respect to the Hausdorff measure \(\mathcal{H}^{n}\) if \(\partial \Omega \) has a sufficiently nice geometry. The main result of the present paper goes in the opposite direction. It says that, if \(E\subset \partial \Omega \) with \(\mathcal{H}^{n}(E)<\infty \), then absolute continuity of \(\omega \) with respect to \(\mathcal{H}^{n}\) on \(E\) implies that \(\omega |_{E}\) is \(n\)-rectifiable, that is, \(\omega \)-almost all of \(E\) can be covered by a countable union of \(n\)-dimensional Lipschitz graphs (suitably rotated). No additional topological conditions are imposed on the domain \(\Omega \). Remarkably, this result is new even when \(n=1\). The proof involves establishing bounds on the Riesz transform of \(\omega \), and then deducing rectifiability using recent work of \textit{F. Nazarov} et al. [Acta Math. 213, No. 2, 237--321 (2014; Zbl 1311.28004); Publ. Mat., Barc. 58, No. 2, 517--532 (2014; Zbl 1312.44005)].