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Harmonic measure of curves in the disk

Authors:	Donald E Marshall Carl Sundberg

Institution:	(1) Mathematical Sciences Research Institute, 1000 Centennial Drive, 94720-5070 Berkeley, CA, USA;(2) Present address: Department of Mathematics, University of Washington, Box 354350, 98195-4350 Seattle, WA, USA;(3) Department of Mathematics, University of Tennessee, 37996-1300 Knoxville, TN, USA

Abstract:	A powerful tool for studying the growth of analytic and harmonic functions is Hall's Lemma, which states that there is a constantC>0 so that the harmonic measure of a subsetE of the closed unit disk $\bar {\mathbb{D}}$ evaluated at 0 satisfies $\omega \left( {0,E,\mathbb{D}\backslash E} \right) \geqslant C_\omega \left( {0,E_{rad} ,\mathbb{D}} \right)$ whereE _rad is the radial projection ofE onto $\partial \mathbb{D}$ . FitzGerald, Rodin and Warschawski proved that ifE is a continuum in $\bar {\mathbb{D}}$ whose radial projection has length at most π then () is true withC=1, and they asked how large the length, \|E _rad\|, can be in order for their result to be valid. We prove that () holds withC=1 for every continuum $E \subset \bar {\mathbb{D}}$ satisfying $\left\| {E_{rad} } \right\| \leqslant \theta _c \simeq 2\pi \left( {\frac{{350}}{{360}}} \right)$ and θc cannot be replaced by a larger number. Fuchs asked for the largest constantC so that () holds for allE. We show that for every continuum $E \subset \bar {\mathbb{D}}$ , () holds withC=C _2π≅.977126698498665669…, whereC _2π is the harmonic measure of the two long sides of a 3∶1 rectangle evaluated at the center. There are Jordan curves for which equality holds in (*) withC=C _2π. The authors are supported in part by NSF grants DMS-9302823 and DMS-9401027, and while at MSRI by NSF grant DMS-9022140.

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