On the snow flake domain
type: Item
instance of: scholarly article
title: On the snow flake domain
zbMATH Open document ID: 0573.30030
DOI: 10.1007/BF02384424
published in: Arkiv för Matematik
publication date: 1985-01-01T00:00:00Z
review text: The Hausdorff dimension of the boundary of the snow flake domain is greater than 1 (it is (log 4)/(log 3)) so it is a natural test case for the conjecture that harmonic measure for a plane domain is always singular with respect to \(\alpha\)-dimensional Hausdorff measure for all \(\alpha >1\) [the reviewer, Pac. J. Math. 95, 179-192 (1981; Zbl 0493.31001)]. In this article the authors prove that there exists a subset of the boundary of the snow flake domain with full harmonic measure and with Hausdorff dimension strictly less than (log 4)/(log 3), thus supporting the conjecture. Subsequently L. Carleson has extended this result and verified the conjecture for all Cantor sets (On the support of the harmonic measure for sets of Cantor type, Mittag-Leffler Institute Report 4, 1984) and recently \textit{N. G. Makarov} has proved that the conjecture holds for all simply connected domains [Lond. Math. Soc. 51, 369-384 (1985; reviewed above)].
Mathematics Subject Classification ID: 30C85
Mathematics Subject Classification ID: 31A15
zbMATH DE Number: 3914674
zbMATH Keywords: snow flake domain
zbMATH Keywords: harmonic measure
zbMATH Keywords: Hausdorff measure
author: Robert Kaufman
author: Jang-Mei Wu
reviewed by: B. Øksendal
MaRDI profile type: Publication
cites work: On the sum of digits of real numbers represented in the dyadic system. (On sets of fractional dimensions II.)
cites work: Boundary behavior of harmonic functions in non-tangentially accessible domains
cites work: Brownian motion and sets of harmonic measure zero
full work available at URL: https://doi.org/10.1007/bf02384424
OpenAlex ID: W1988297186
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description: scientific article; zbMATH DE number 3914674