On conformal mapping and linear measure
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.
scientific article; zbMATH DE number 3975314
scientific article