Hausdorff dimension of harmonic measures in the plane
The main result of this paper is the following great theorem: Theorem 1. Let \(\Omega\) be a planar domain whose complement has positive (logarithmic) capacity. Then there is a set \(F\subset \partial \Omega\) with Hausdorff dimension \(\leq 1\) such that \(\omega (\Omega,F,z)=1\) for all \(z\in \Omega\), where \(\omega\) (\(\Omega\),F,z) denotes the harmonic measure of F with respect to \(\Omega\) and \(z\in \Omega\). This theorem gives a complete affirmative answer to a conjecture by \textit{B. Øksendal [Pac. J. Math. 95, 179-192 (1981; Zbl 0493.31001)]. Before the present work special cases of this conjecture had been established by several authors, among them L. Carleson (assuming a certain scale invariance) and N. G. Makarov (for simply connected domains). The proof of Theorem 1 is based on an ingenious combination of ideas from Makarov's and Carleson's works. It uses a domain modification procedure due to Carleson which simplified considerably the argument originally found by the authors.}
scientific article; zbMATH DE number 4091817