Hausdorff dimension of harmonic measures in the plane
type: Item
instance of: scholarly article
title: Hausdorff dimension of harmonic measures in the plane
zbMATH Open document ID: 0667.30020
DOI: 10.1007/BF02392296
author: Peter Jones
author: Thomas H. Wolff
published in: Acta Mathematica
publication date: 1988-01-01T00:00:00Z
review text: 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.}
Mathematics Subject Classification ID: 30C85
Mathematics Subject Classification ID: 31A15
zbMATH DE Number: 4091817
zbMATH Keywords: harmonic measure
zbMATH Keywords: Hausdorff measure
zbMATH Keywords: Green function
zbMATH Keywords: level curves
zbMATH Keywords: critical points
reviewed by: B. Øksendal
MaRDI profile type: Publication
cites work: Negatively curved manifolds, elliptic operators, and the Martin boundary
cites work: On the Hausdorff dimension of harmonic measure in higher dimension
cites work: Can one hear the dimension of a fractal?
cites work: On the support of harmonic measure for sets of Cantor type
cites work: Critical points of Green's function, harmonic measure, and the corona problem
cites work: On the snow flake domain
cites work: On the Distortion of Boundary Sets Under Conformal Mappings
cites work: The dimension of the maximal measure for a polynomial map
cites work: Null Sets for Measures Orthogonal to R(X)
cites work: Brownian motion and sets of harmonic measure zero
cites work: Q3336014
cites work: Hausdorff dimension of harmonic measure on the boundary of an attractive basin for a holomorphic map
full work available at URL: https://doi.org/10.1007/bf02392296
OpenAlex ID: W2084462739
description: scientific article; zbMATH DE number 4091817