Sets of Absolute Continuity for Harmonic Measure in NTA Domains

We show that if Ω is an NTA domain with harmonic measure ω and E⊆∂Ω is contained in an Ahlfors regular set, then ω|E≪ℋd|E\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\omega |_{E}\ll \mathcal {H}^{d}|_{E}$\end{document}. Moreover, this holds quantitatively in the sense that for all τ>0ω obeys an A∞-type condition with respect to ℋd|E′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {H}^{d}|_{E^{\prime }}$\end{document}, where E′⊆E is so that ω(E∖E′)<τω(E), even though ∂Ω may not even be locally ℋd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {H}^{d}$\end{document}-finite. We also show that, for uniform domains with uniform complements, if E⊆∂Ω is the Lipschitz image of a subset of ℝd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {R}^{d}$\end{document}, then there is E′⊆E with ℋd(E∖E′)<τℋd(E)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {H}^{d}(E\backslash E^{\prime })<\tau \mathcal {H}^{d}(E)$\end{document} upon which a similar A∞-type condition holds.