# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.