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Harmonic measure and approximation of uniformly rectifiable sets. (English) Zbl 1371.28002
Let \(E\subset \mathbb{R}^{n+1}\) be a uniformly rectifiable set of dimension \(n\). In this paper, it is proved that for each “dyadic cube” \(Q\) of \(E\) (in the sense of David-Semmes and Christ), there exists an open set \(\widetilde{\Omega}\subset \mathbb{R}^{n+1}\setminus E\) with \(\text{diam}(\widetilde{\Omega}) \approx \text{diam}(Q)\) such that \(\widetilde{\Omega}\) has an Ahlfors-David regular boundary, satisfies a 2-sided corkscrew condition and \(\sigma(\partial \widetilde{\Omega}\cap Q) \gtrsim \sigma(Q)\), where \(\sigma=H^n|_E\) is the surface measure on \(E\) and \(H^n\) the \(n\)-dimensional Hausdorff measure; moreover, each connected component of \(\widetilde{\Omega}\) is a NTA (non-tangentially accessible) domain with Ahlfors-David regular boundary. That is, \(E\) has big pieces of boundaries of a class of domains which satisfy a 2-sided corkscrew condition, and whose connected components are all chord-arc domains.
Furthermore, \(E\) also has “interior big pieces of good harmonic measure estimates” in the sense that the set \(\widetilde{\Omega}\) related to \(Q\) also satisfies that, for each surface ball \(\Delta:=\Delta(x):=B(x,r)\cap \partial \widetilde{\Omega}\), with \(x\in \partial \widetilde{\Omega}\) and \(r\in(0,\text{diam}(\widetilde{\Omega}))\), and with interior corkscrew point \(X_\Delta\), the harmonic measure for \(\widetilde{\Omega}\) with pole at \(X_\Delta\) belongs to weak-\(A_\infty(\Delta)\).
Reviewer: Wen Yuan (Beijing)

MSC:
28A75 Length, area, volume, other geometric measure theory
28A78 Hausdorff and packing measures
31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
42B25 Maximal functions, Littlewood-Paley theory
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