On conformal mapping and linear measure
type: Item
instance of: scholarly article
title: On conformal mapping and linear measure
zbMATH Open document ID: 0604.30029
DOI: 10.1007/BF02796587
published in: Journal d'Analyse Mathématique
publication date: 1986-01-01T00:00:00Z
review text: Recently \textit{N. G. Makarov} [Proc. Lond. Math. Soc. 51, 369-384 (1985; Zbl 0573.30029)] proved that if f is a conformal map defined on the unit disc D in the complex plane, and h(t) is a strictly increasing real function such that \(h(0)=0\) and \(t^{-1} h(t)\to 0\) as \(t\to 0^+\), then there exists a set \(A\subset \partial D\) of length \(2\pi\) such that \(\Lambda_ h(f(A))=0\) (where \(\Lambda_ h\) denotes the Hausdorff measure associated to h), thereby solving a conjecture of Øksendal (1981). In this article some of Makorov's ideas together with McMillan's twist point theorem and Plessner's theorem are used to obtain the following even stronger result: There exists a set \(A\subset \partial D\) of length \(2\pi\) such that \(f(A)\) has \(\sigma\)-finite length. Moreover, f is absolutely continuous on A.
Mathematics Subject Classification ID: 30C85
Mathematics Subject Classification ID: 30E25
Mathematics Subject Classification ID: 31A15
zbMATH DE Number: 3975314
zbMATH Keywords: conformal mapping
zbMATH Keywords: harmonic measure
zbMATH Keywords: Hausdorff measure
zbMATH Keywords: McMillan's twist point theorem
zbMATH Keywords: Plessner's theorem
author: Christian Pommerenke
reviewed by: B. Øksendal
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02796587
OpenAlex ID: W4230274343
description: scientific article; zbMATH DE number 3975314