Sets of Absolute Continuity for Harmonic Measure in NTA Domains

Jonas Azzam

Research output: Contribution to journalArticlepeer-review

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 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.
Original languageEnglish
Pages (from-to)403-433
Number of pages31
JournalPotential analysis
Volume45
Issue number3
Early online date18 Mar 2016
DOIs
Publication statusPublished - Oct 2016
Externally publishedYes

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