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On the distortion of boundary sets under conformal mappings. (English) Zbl 0573.30029
In this impressive paper the following three spectacular results are proved: Theorem 1: There exists a universal constant $$C>0$$ such that for any Jordan domain $$\Omega$$ the harmonic measure $$\omega$$ on $$\partial \Omega$$ is absolutely continuous with respect to the Hausdorff measure $$\Lambda_{\phi}$$ with $\phi (t)=t \exp \{C(\log (1/t)\log \log \log (1/t))^{1/2}\}.$ Theorem 2: There exists a $$c>0$$ and a Jordan domain $$\Omega$$ such that $$\omega$$ is singular with respect to $$\Lambda_{\phi}$$ if $\phi (t)=t \exp \{c(\log (1/t)\log \log \log (1/t))^{1/2}\}.$ Theorem 3: For any simply connected domain $$\Omega$$ and for any measure function $$\phi$$ such that $$\lim_{t\to 0}(\phi (t)/t)=0$$, $$\omega$$ is singular with respect to $$\Lambda_{\phi}$$. Theorem 1 extends a result due to L. Carleson [Duke Math. J. 40, 547-559 (1973; Zbl 0273.30014)], theorem 2 refines a result due to R. Kaufman and J.-M. Wu [Mich. Math. J. 29, 267-280 (1982; Zbl 0538.30019)] and theorem 3 answers in affirmative a conjecture by the reviewer [Pac. J. Math. 95, 179-192 (1981; Zbl 0493.31001)] in the simply connected case.
Reviewer: B.Øksendal

##### MSC:
 30C85 Capacity and harmonic measure in the complex plane 31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
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