On the distortion of boundary sets under conformal mappings. (English) Zbl 0573.30029

Makarov, N. G.

Proc. Lond. Math. Soc., III. Ser. 51, 369-384 (1985).

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

Keywords:

iterated logarithm for Bloch functions; harmonic measure; Hausdorff measure; Jordan domain

Citations:

Zbl 0573.30030; Zbl 0273.30014; Zbl 0538.30019; Zbl 0493.31001