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The planar Cantor sets of zero analytic capacity and the local \(T(b)\)-Theorem. (English) Zbl 1016.30020
The analytic capacity of a compact set \(E\) in the complex plane \(\mathbb C\) is \(\gamma(E)=\sup |f^\prime(\infty)|\), where the supremum is taken over all analytic functions \(f\) on \(\mathbb C\setminus E\) such that \(|f|\leq 1\). Analytic capacity has recently been the object of extensive study and there has been important progress on our understanding of it. The authors prove two theorems for the analytic capacity of planar Cantor sets. If \(\lambda=\{\lambda_n\}\) is a sequence with \(0\leq \lambda_n\leq 1/2\), let \(E=E(\lambda)\) be the Cantor set constructed inductively inside the unit square in the usual way. Let \(E_n\) be the \(n\)th approximation of \(E\), that is the set constructed at the \(n\)th step (\(E=\bigcap_{n=1}^\infty E_n\)). Theorem 1 says that \(\gamma(E)=0\) if and only if \[ \sum_{n=1}^\infty \frac{1}{(4^n \lambda_1\lambda_2\dots\lambda_n)^2}=\infty. \] This result answers a question of P. Mattila [Publ. Mat. 40, 195-204 (1996; Zbl 0888.30026)] and completes the solution of a long-standing open problem; special cases of it had been solved by various authors. Theorem 1 follows from Theorem 2 which concerns a related quantity, the positive analytic capacity of Cantor sets, and confirms a conjecture of V. Y. Eiderman [Math. Notes 63, 813-822 (1998; Zbl 0919.28004)]. It implies that \(\gamma(E_n)\leq Cn^{-1/2}\), \(n=1,2,\dots\). The proof of Theorem 2 is technical; its main tool is M. Christ’s \(T(b)\)-Theorem. The second author, X. Tolsa, has recently proved important results that extend the results of the present article; see Painlevé’s problem and the semiadditivity of analytic capacity to appear in Acta Math..

MSC:
30C85 Capacity and harmonic measure in the complex plane
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
30E20 Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane
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