Wolff snowflakes
type: Item
instance of: scholarly article
title: Wolff snowflakes
zbMATH Open document ID: 1108.31006
author: John L. Lewis
author: Gregory Verchota
author: Andrew Vogel
published in: Pacific Journal of Mathematics
publication date: 2007-01-19T00:00:00Z
review text: Let \(D\) be a domain in \(\mathbb R^n\) (\(n\geq 3\)) and \(\mu\) be harmonic measure for \(D\) and some fixed point. Further, let \(\text{H-dim\,}\mu\) denote the infimum of those values \(k>0\) for which there exists \(E\subset \partial D\) with zero \(k\)-dimensional Hausdorff measure and \(\mu(E)=1\). \textit{T. H. Wolff} [Essays on Fourier analysis in honor of Elias M. Stein. Princeton Math. Ser. 42, 321--384 (1995; Zbl 0836.31004)] constructed certain domains \(D\) in \(\mathbb R^3\) with fractal-like boundaries, now called ``Wolff snowflakes'', for which either \(\text{H-dim\,}\mu >2\) or \(\text{H-dim}\;\mu <2 \). Let \(\widetilde{\mu }\) denote harmonic measure for \(\mathbb R^n\setminus\overline{D}\) and some fixed point. Then Wolff's construction actually leads to snowflakes in \(\mathbb R^n\) for which \(\text{H-dim }\mu >n-1\) and \(\text{H-dim}\;\widetilde{\mu}<n-1\); and these inequalities can be simultaneously reversed. The present paper verifies a conjecture of Wolff, that there are snowflakes for which \(\text{H-dim}\;\mu >n-1\) and \(\text{H-dim}\;\widetilde{\mu }>n-1\); and these inequalities can also be simultaneously reversed. The proofs are based on extensive difficult computations.
Mathematics Subject Classification ID: 31B05
Mathematics Subject Classification ID: 28A80
Mathematics Subject Classification ID: 31B15
zbMATH DE Number: 5117285
zbMATH Keywords: harmonic measure
zbMATH Keywords: Green function
zbMATH Keywords: Hausdorff measure
zbMATH Keywords: harmonic function
reviewed by: Stephen J. Gardiner
MaRDI profile type: Publication
full work available at URL: pjm.2005.218.139
OpenAlex ID: W4254533365
DOI: 10.2140/PJM.2005.218.139
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description: scientific article; zbMATH DE number 5117285