Type: Article
Publication Date: 2019-01-01
Citations: 12
DOI: https://doi.org/10.4310/arkiv.2019.v57.n1.a1
We prove a structure theorem for any n-rectifiable set E ⊂ R n+1 , n ≥ 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 R n+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 H n | 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 R n+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 ⊂ R n+1 which satisfies an infinitesimal interior thickness condition, then H n | ∂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 [AHM 3 TV], 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.