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We study the properties of harmonic measure in semi-uniform domains. Aikawa and Hirata showed in [AH08] that, for John domains satisfying the capacity density condition (CDC), the doubling property for harmonic measure is equivalent to the domain being semi-uniform. Our first result removes the John condition by showing that any domain satisfying the CDC whose harmonic measure is doubling is semiuniform. Next, we develop a substitute for some classical estimates on harmonic measure in nontangentially accessible domains that works in semi-uniform domains. We also show that semi-uniform domains with uniformly rectifiable boundary have big pieces of chord-arc subdomains. We cannot hope for big pieces of Lipschitz subdomains (as was shown for chord-arc domains by David and Jerison [DJ90]) due to an example of Hrycak, which we review in the appendix. Finally, we combine these tools to study the A-property of harmonic measure. For a domain with Ahlfors-David regular boundary, it was shown by Hofmann and Martell that the A property of harmonic measure implies uniform rectifiability of the boundary [HM15, HLMN17]. Since A∞-weights are doubling, this also implies the domain is semi-uniform. Our final result shows that these two properties, semi-uniformity and uniformly rectifiable boundary, also imply the Aproperty for harmonic measure, thus classifying geometrically all domains for which this holds.
Original languageEnglish
Number of pages55
JournalInternational Mathematics Research Notices
Early online date5 Mar 2019
Publication statusE-pub ahead of print - 5 Mar 2019

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