Tangent measures and absolute continuity of harmonic measure (Q1709562)

This paper investigates when the harmonic measure \(\omega _{\Omega }\) for a domain \(\Omega \) in \(\mathbb{R}^{d+1}\), \(d\geq 2\), can be mutually absolutely continuous with respect to \(\alpha \)-dimensional Hausdorff measure \(\mathcal{H}^{\alpha }\), where \(\alpha \) may differ from \(d\). Two sample corollaries of the main result will give the flavour. Corollary I. If \(\Omega \) is an NTA domain and \(E\subset \mathbb{R}^{d+1}\) satisfies \(\omega _{\Omega }(E)>0\) and \(\omega _{\Omega }|_{E}\ll \) \(\mathcal{H}^{\alpha }|_{E}\ll \omega _{\Omega }|_{E}\), then \(\alpha \leq d\). Corollary III. Let \(\Omega \) be a uniform domain, \(E\) be a set of ``non-degenerate'' points of \(\partial \Omega \), and \(\alpha >d\). If \(\omega_{\Omega }|_{E}\ll \) \(\mathcal{H}^{\alpha }|_{E}\ll \omega _{\Omega }|_{E}\), then \(\omega _{\Omega }(E)=0\).