Large sets of zero analytic capacity (Q2750849)
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scientific article; zbMATH DE number 1663094
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Large sets of zero analytic capacity |
scientific article; zbMATH DE number 1663094 |
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Large sets of zero analytic capacity (English)
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21 October 2001
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analytic capacity
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Hausdorff measure
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Cantor set
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harmonic measure
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The authors consider a Cantor type set in the unit square and prove that it has zero analytic capacity but it does not have \(\sigma\)-finite one-dimensional Hausdorff measure. The first example of such a Cantor set was given by L. D. Ivanov [On sets of analytic capacity zero, in: Linear and Complex Analysis Problem Book 3, Part II (V.P. Khavin, S.V. Kruschev, N.K. Nikolskii, ed.), Lecture Notes Math. 1043, 498--501 (1984; Zbl 0545.30038)]. The proof in the paper under review is based on some estimates for harmonic measure and Green function due to P. W. Jones [Lecture Notes Math. 1384, 24--68 (1989; Zbl 0675.30029)] and a stopping-time argument. The result is related to a conjecture of P. Mattila [Publ. Math., Barc. 40, No. 1, 195--204 (1996; Zbl 0888.30026)] on the characterization of Cantor sets with zero analytic capacity.
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