Dimension gap under conformal mappings
type: Item
instance of: scholarly article
title: Dimension gap under conformal mappings
zbMATH Open document ID: 1253.30034
author: Stanislav Hencl
author: Pekka Koskela
author: Tomi Nieminen
published in: Advances in Mathematics
publication date: 2012-07-20T00:00:00Z
Mathematics Subject Classification ID: 30C65
Mathematics Subject Classification ID: 28A78
zbMATH DE Number: 6059034
zbMATH Keywords: conformal mapping
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: boundary behaviour
reviewed by: Vesna Manojlovic
MaRDI profile type: Publication
full work available at URL: j.aim.2012.03.018
OpenAlex ID: W1978059905
Wikidata QID: Q110124878
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DOI: 10.1016/J.AIM.2012.03.018
review text: Let \(f : \Delta \rightarrow \Omega\) be a conformal map of the unit disc \(\Delta\) onto a simply connected domain \(\Omega\) and let \(p > 1\). The main result of this paper is the following theorem:NEWLINENEWLINEIf \(\int_\Delta |f'(z)|^2 \log^p (1-|z|)^{-1} dA < \infty\), then \(\psi\)-Hausdorff dimension of \(\partial \Omega\) is zero, i.e., \(H^{\psi}(\partial \Omega) = 0\), for the gauge function \(\psi(t) = t^2 (-\log t)^s\) and \(s < 2p\).NEWLINENEWLINEThis result is extended to multiply connected domains satisfying \(k_{\Omega}(z_0, \cdot) \in L^p(\Omega)\), where \(k_\Omega\) is the quasihyperbolic metric in \(\Omega\).NEWLINENEWLINEThese results are essentially sharp as is shown by constructing an example. The authors make an important point that one can have \(H^\psi (\partial \Omega) = \infty\) for all \(s>0\) with \(\psi(t) = t^2 (-\log t)^s\) if one only assumes \(|f^\prime| \in L^2(\Delta, |\log (1-|z|)| dA(z))\). Therefore, there is a dimension gap as \(p \to 1\).
description: scientific article; zbMATH DE number 6059034