Sets of Absolute Continuity for Harmonic Measure in NTA Domains

Abstract

We show that if Ω is an NTA domain with harmonic measure ω and E⊆∂Ω is contained in an Ahlfors regular set, then $\omega |_{E}\ll \mathcal {H}^{d}|_{E}$. Moreover, this holds quantitatively in the sense that for all τ>0ω obeys an A _∞ -type condition with respect to $\mathcal {H}^{d}|_{E^{\prime }}$, where E ^′⊆E is so that ω(E∖E ^′)<τ ω(E), even though ∂Ω may not even be locally $\mathcal {H}^{d}$-finite. We also show that, for uniform domains with uniform complements, if E⊆∂Ω is the Lipschitz image of a subset of $\mathbb {R}^{d}$, then there is E ^′⊆E with $\mathcal {H}^{d}(E\backslash E^{\prime })<\tau \mathcal {H}^{d}(E)$ upon which a similar A _∞ -type condition holds.