Rectifiability, interior approximation and harmonic measure

Murat Akman University of Connecticut Simon Bortz University of Washington Steve Hofmann University of Missouri José María Martell Instituto de Ciencias Matemáticas, Madrid, Spain

Classical Analysis and ODEs mathscidoc:1912.43026

Arkiv for Matematik, 57, (1), 1 – 22, 2019
We prove a structure theorem for any n-rectifiable set E⊂Rn+1,n≥1, satisfying a weak version of the lower ADR condition, and having locally finite Hn (n-dimensional Hausdorff) measure. Namely, that Hn-almost all of E can be covered by a countable union of boundaries of bounded Lipschitz domains contained in Rn+1∖E. As a consequence, for harmonic measure in the complement of such a set E, we establish a non-degeneracy condition which amounts to saying that Hn|E is “absolutely continuous” with respect to harmonic measure in the sense that any Borel subset of E with strictly positive Hn measure has strictly positive harmonic measure in some connected component of Rn+1∖E. We also provide some counterexamples showing that our result for harmonic measure is optimal. Moreover, we show that if, in addition, a set E as above is the boundary of a connected domain Ω⊂Rn+1 which satisfies an infinitesimal interior thickness condition, then Hn|∂Ω is absolutely continuous (in the usual sense) with respect to harmonic measure for Ω. Local versions of these results are also proved: if just some piece of the boundary is n-rectifiable then we get the corresponding absolute continuity on that piece. As a consequence of this and recent results in “Rectifiability of harmonic measure” [Geom. Funct. Anal. 26 (2016), 703–728], we can decompose the boundary of any open connected set satisfying the previous conditions in two disjoint pieces: one that is n-rectifiable where Hausdorff measure is absolutely continuous with respect to harmonic measure and another purely n-unrectifiable piece having vanishing harmonic measure.
harmonic measure, rectifiability
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@inproceedings{murat2019rectifiability,,
  title={Rectifiability, interior approximation and harmonic measure},
  author={Murat Akman, Simon Bortz, Steve Hofmann, and José María Martell},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191204132421094298582},
  booktitle={Arkiv for Matematik},
  volume={57},
  number={1},
  pages={1 – 22},
  year={2019},
}
Murat Akman, Simon Bortz, Steve Hofmann, and José María Martell. Rectifiability, interior approximation and harmonic measure. 2019. Vol. 57. In Arkiv for Matematik. pp.1 – 22. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191204132421094298582.
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