On the snow flake domain. (English) Zbl 0573.30030

Kaufman, Robert; Wu, Jang-Mei

Ark. Mat. 23, 177-183 (1985).

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

MSC:

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

Keywords:

snow flake domain; harmonic measure; Hausdorff measure

Citations:

Zbl 0573.30029; Zbl 0493.31001

References:

[1]	Besicovitch, A. S., On the sum of digits of real numbers represented in the dyadic system, (On sets of fractional dimensions II),Math. Annalen 110 (1934–35), 321–330. · Zbl 0009.39503
[2]	Jerison, D. S. andKenig, C. E., Boundary behaviour of harmonic functions in non-tangentially accessible domains,Adv. Maths. 46 (1982), 80–147. · Zbl 0514.31003
[3]	Lehto, O. andVirtanen, K. O.,Quasiconformal Mappings in the Plane, Springer-Verlag (1973).
[4]	Øksendal, B., Brownian motion and sets of harmonic measure zero,Pacific J. Math. 95 (1981). 193–204. · Zbl 0493.31001