Arkiv för Matematik

Volume 57 (2019)

Number 1

Rectifiability, interior approximation and harmonic measure

Pages: 1 – 22

DOI: https://dx.doi.org/10.4310/ARKIV.2019.v57.n1.a1

Authors

Murat Akman (Department of Mathematics, University of Connecticut, Storrs, Ct., U.S.A.)

Simon Bortz (Department of Mathematics, University of Washington, Seattle, Wa., U.S.A.)

Steve Hofmann (Department of Mathematics, University of Missouri, Columbia, Mo., U.S.A.)

José María Martell (Consejo Superior de Investigaciones Científicas, Instituto de Ciencias Matemáticas, Madrid, Spain)

Abstract

We prove a structure theorem for any $n$-rectifiable set $E \subset \mathbb{R}^{n+1}, n \geq 1$, satisfying a weak version of the lower ADR condition, and having locally finite $H^n$ ($n$-dimensional Hausdorff) measure. Namely, that $H^n$-almost all of $E$ can be covered by a countable union of boundaries of bounded Lipschitz domains contained in $\mathbb{R}^{n+1} \setminus E$. 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 $H^n \vert {}_E$ is “absolutely continuous” with respect to harmonic measure in the sense that any Borel subset of $E$ with strictly positive $H^n$ measure has strictly positive harmonic measure in some connected component of $\mathbb{R}^{n+1} \setminus E$. 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 $\Omega \subset \mathbb{R}^{n+1}$ which satisfies an infinitesimal interior thickness condition, then $H^n \vert {}_{\partial \Omega}$ is absolutely continuous (in the usual sense) with respect to harmonic measure for $\Omega$. 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.

Keywords

harmonic measure, rectifiability

2010 Mathematics Subject Classification

28A75, 28A78, 30C85, 31A15, 31B05, 42B37, 49Q15

Full Text (PDF format)

The first and last authors acknowledge financial support from the Spanish Ministry of Economy and Competitiveness, through the “Severo Ochoa” Programme for Centres of Excellence in R&D” (SEV-2015-0554). They also acknowledge that the research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ ERC agreement no. 615112 HAPDEGMT. The second and third authors were supported by NSF grant DMS-1361701. The last author would like to express his gratitude to the University of Missouri-Columbia (USA), for its support and hospitality while he was visiting this institution. All authors wish to thank Matthew Badger, Svitlana Mayboroda, and Tatiana Toro for their helpful comments and suggestions.

Received 31 May 2017

Accepted 8 October 2018

Published 3 May 2019