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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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##### References:
 [1] Michael Christ, Lectures on singular integral operators, CBMS Regional Conference Series in Mathematics, vol. 77, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1990. · Zbl 0745.42008 [2] Michael Christ, A \?(\?) theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math. 60/61 (1990), no. 2, 601 – 628. · Zbl 0758.42009 [3] Guy David, Analytic capacity, Calderón-Zygmund operators, and rectifiability, Publ. Mat. 43 (1999), no. 1, 3 – 25. · Zbl 1049.30017 · doi:10.5565/PUBLMAT_43199_01 · doi.org [4] Alexander M. Davie and Bernt Øksendal, Analytic capacity and differentiability properties of finely harmonic functions, Acta Math. 149 (1982), no. 1-2, 127 – 152. · Zbl 0527.31001 · doi:10.1007/BF02392352 · doi.org [5] V. Ya. Èĭderman, Hausdorff measure and capacity associated with Cauchy potentials, Mat. Zametki 63 (1998), no. 6, 923 – 934 (Russian, with Russian summary); English transl., Math. Notes 63 (1998), no. 5-6, 813 – 822. · Zbl 0919.28004 · doi:10.1007/BF02312776 · doi.org [6] John Garnett, Positive length but zero analytic capacity, Proc. Amer. Math. Soc. 24 (1970), 696-699; errata, ibid. 26 (1970), 701. · Zbl 0208.09803 [7] John Garnett, Analytic capacity and measure, Lecture Notes in Mathematics, Vol. 297, Springer-Verlag, Berlin-New York, 1972. · Zbl 0253.30014 [8] J. GARNETT, S. YOSHINOBU, Large sets of zero analytic capacity, Proc. Amer. Math. Soc. 129 (2001), 3543-3548. · Zbl 1062.30025 [9] Вариации множеств и функций., Издат. ”Наука”, Мосцощ, 1975 (Руссиан). Едитед бы А. Г. Витуšкин. [10] L. D. IVANOV, On sets of analytic capacity zero, in “Linear and Complex Analysis Problem Book 3” (part II), Lectures Notes in Mathematics 1574, Springer-Verlag, Berlin, 1994, pp. 150-153. [11] Peter W. Jones, Square functions, Cauchy integrals, analytic capacity, and harmonic measure, Harmonic analysis and partial differential equations (El Escorial, 1987) Lecture Notes in Math., vol. 1384, Springer, Berlin, 1989, pp. 24 – 68. · doi:10.1007/BFb0086793 · doi.org [12] Pertti Mattila, On the analytic capacity and curvature of some Cantor sets with non-\?-finite length, Publ. Mat. 40 (1996), no. 1, 195 – 204. · Zbl 0888.30026 · doi:10.5565/PUBLMAT_40196_12 · doi.org [13] Takafumi Murai, Construction of \?\textonesuperior functions concerning the estimate of analytic capacity, Bull. London Math. Soc. 19 (1987), no. 2, 154 – 160. · Zbl 0634.30025 · doi:10.1112/blms/19.2.154 · doi.org [14] Xavier Tolsa, \?²-boundedness of the Cauchy integral operator for continuous measures, Duke Math. J. 98 (1999), no. 2, 269 – 304. · Zbl 0945.30032 · doi:10.1215/S0012-7094-99-09808-3 · doi.org [15] X. TOLSA, On the analytic capacity $$\gamma^+$$, Indiana Univ. Math. J. (51) (2) (2002), 317-344. [16] X. TOLSA, Painlevé’s problem and the semiadditivity of analytic capacity, Acta Math. (to appear). · Zbl 1060.30031 [17] N. X. UY, Removable sets of analytic functions satisfying a Lipschitz condition, Ark. Mat. 17 (1979), 19-27. · Zbl 0442.30033 [18] Joan Verdera, A weak type inequality for Cauchy transforms of finite measures, Publ. Mat. 36 (1992), no. 2B, 1029 – 1034 (1993). · Zbl 0776.30028 · doi:10.5565/PUBLMAT_362B92_19 · doi.org [19] Joan Verdera, Removability, capacity and approximation, Complex potential theory (Montreal, PQ, 1993) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 439, Kluwer Acad. Publ., Dordrecht, 1994, pp. 419 – 473. · Zbl 0809.30001 [20] Joan Verdera, \?² boundedness of the Cauchy integral and Menger curvature, Harmonic analysis and boundary value problems (Fayetteville, AR, 2000) Contemp. Math., vol. 277, Amer. Math. Soc., Providence, RI, 2001, pp. 139 – 158. · Zbl 1002.42011 · doi:10.1090/conm/277/04543 · doi.org [21] A. G. VITUSHKIN, Estimate of the Cauchy integral, Mat. Sb. 71(4) (1966), 515-534. [22] A. G. VITUSHKIN, Analytic capacity of sets in problems of approximation theory, Russian Math. Surveys 22 (1967), 139-200. · Zbl 0164.37701
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