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Counterexamples with harmonic gradients in \(\mathbb{R}^ 3\). (English) Zbl 0836.31004

Fefferman, Charles (ed.) et al., Essays on Fourier analysis in honor of Elias M. Stein. Proceedings of the Princeton conference on harmonic analysis held at Princeton Univ., Princeton, NJ, USA, May 13-17, 1991 in honor of Elias M. Stein’s 60th birthday. Princeton, NJ: Princeton Univ. Press. Princeton Math. Ser. 42, 321-384 (1995).
P. W. Jones and the author [Acta Math. 161, 131-144 (1988; Zbl 0667.30020)]have proved that the harmonic measure of any domain in \(\mathbb{R}^2\) puts full mass on some set with Hausdorff dimension 1. In the present paper it is shown that an analog in higher dimensions is not valid. Let \(\alpha >0\) be small enough. There is a bounded domain in \(\mathbb{R}^3\) whose harmonic measure puts no mass on any set with Hausdorff dimension less than \(2+ \alpha\). Two more interesting counterexamples are also given.
For the entire collection see [Zbl 0810.00019].
Reviewer: M.Dont (Praha)

MSC:

31B15 Potentials and capacities, extremal length and related notions in higher dimensions
30C85 Capacity and harmonic measure in the complex plane
31B20 Boundary value and inverse problems for harmonic functions in higher dimensions

Citations:

Zbl 0667.30020
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