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The planar Cantor sets of zero analytic capacity and the local $T(b)$-Theorem


Authors: Joan Mateu, Xavier Tolsa and Joan Verdera
Journal: J. Amer. Math. Soc. 16 (2003), 19-28
MSC (2000): Primary 30C85; Secondary 42B20, 30E20
DOI: https://doi.org/10.1090/S0894-0347-02-00401-0
Published electronically: July 10, 2002
MathSciNet review: 1937197
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Abstract: In this paper we obtain rather precise estimates for the analytic capacity of a big class of planar Cantors sets. In fact, we show that analytic capacity and positive analytic capacity are comparable for these sets. The main tool for the proof is an appropriate version of the $T(b)$-Theorem.


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Additional Information

Joan Mateu
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, (Barcelona), Spain

Xavier Tolsa
Affiliation: Département de Mathématiques, Université de Paris Sud 91405 Orsay, cedex, France

Joan Verdera
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, (Barcelona), Spain

DOI: https://doi.org/10.1090/S0894-0347-02-00401-0
Keywords: Analytic capacity, Cauchy integral, Cantor sets, $T(b)$-Theorem, positive analytic capacity
Received by editor(s): August 7, 2001
Published electronically: July 10, 2002
Additional Notes: The authors were partially supported by the grants BFM 2000-0361, HPRN-2000-0116 and 2001- SGR-00431. The second author was supported by a Marie Curie Fellowship of the European Union under contract HPMFCT-2000-00519.
Article copyright: © Copyright 2002 American Mathematical Society

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