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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)

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
Full Text: DOI arXiv
[1] Auscher, P., Hofmann, S., Lewis, J. L. and Tchamitchian, P.: Extrapolation of Carleson measures and the analyticity of Kato’s square-root operators. Acta Math.187 (2001), no. 2, 161–190. · Zbl 1163.35346
[2] Auscher, P., Hofmann, S., Muscalu, C., Tao, T. and Thiele, C.: Carleson measures, trees, extrapolation, andT (b) theorems. Publ. Mat. 46 (2002), no. 2, 257–325. · Zbl 1027.42009
[3] Azzam, J., Hofmann, S., Martell, J. M., Nystr”om, K. and Toro, T.:A new characterization of chord-arc domains. To appear in J. Euro. Math. Soc. · Zbl 1366.28004
[4] Azzam, J. and Schul, R.: Hard Sard: quantitative implicit function and extension theorem for Lipschitz maps. Geom. Funct. Anal.22 (2012), no. 5, 1062–1123. · Zbl 1271.26004
[5] Bishop, C. and Jones, P.: Harmonic measure and arclength. Ann. of Math. (2)132 (1990), no. 3, 511–547. · Zbl 0726.30019
[6] Bennewitz, B. and Lewis, J.L.: On weak reverse H”older inequalities for nondoubling harmonic measures. Complex Var. Theory Appl.49 (2004), no. 7-9, 571–582. · Zbl 1068.31001
[7] Carleson, L.: Interpolations by bounded analytic functions and the corona prob372S. Bortz and S. Hofmann · Zbl 0192.16801
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