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Content and harmonic measure: An extension of Hall’s lemma. (English) Zbl 0639.31004
Let D be a domain in \({\mathbb{R}}^ m,\) \(m\geq 2\), satisfying certain conditions, and let P be a fixed point of D. The paper contains estimates for the harmonic measure at P of a closed subset E of D, relative to \(D\setminus E\). A lower estimate (in which E need be only relatively closed) is given in terms of the m-1 dimensional content of E, which generalizes Hall’s lemma. Both upper and lower estimates are given in terms of the supremum of a class of measures on E, extending a result of L. Carleson [Ann. Acad. Sci. Fenn., Ser. A I 7, 25-32 (1982; Zbl 0521.30026)].
Reviewer: N.A.Watson

31B15 Potentials and capacities, extremal length and related notions in higher dimensions
28A12 Contents, measures, outer measures, capacities
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