On the snow flake domain. (English) Zbl 0573.30030

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 N. G. Makarov has proved that the conjecture holds for all simply connected domains [Lond. Math. Soc. 51, 369-384 (1985; reviewed above)].
Reviewer: B.Øksendal


30C85 Capacity and harmonic measure in the complex plane
31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
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