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. Moreover, this holds quantitatively in the sense that for all τ>0ω obeys an A∞-type condition with respect to Hd|E′, where E′⊆E is so that ω(E∖E′)<τω(E), even though ∂Ω may not even be locally Hd-finite. We also show that, for uniform domains with uniform complements, if E⊆∂Ω is the Lipschitz image of a subset of Rd, then there is E′⊆E with Hd(E∖E′)<τHd(E) upon which a similar A∞-type condition holds.