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Painlevé's problem and the semiadditivity of analytic capacity

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References

  1. Ahlfors, L. V., Bounded analytic functions.Duke Math. J., 14 (1947), 1–11.

    Article  MATH  MathSciNet  Google Scholar 

  2. Christ, M., AT (b) theorem with remarks on analytic capacity and the Cauchy integral.Colloq. Math. 60/61 (1990), 601–628.

    MathSciNet  Google Scholar 

  3. David, G.,Wavelets and Singular Integrals on Curves and Surfaces. Lecture Notes in Math., 1465. Springer-Verlag, Berlin, 1991.

    Google Scholar 

  4. — Unrectifiable 1-sets have vanishing analytic capacity.Rev. Mat. Iberoamericana, 14 (1998), 369–479.

    MATH  MathSciNet  Google Scholar 

  5. — Analytic capacity, Calderón-Zygmund operators, and rectificability.Publ. Mat., 43 (1999), 3–25.

    MATH  MathSciNet  Google Scholar 

  6. David, G. &Mattila, P., Removable sets for Lipschitz harmonic functions in the plane.Rev. Mat. Iberoamericana, 16 (2000), 137–215.

    MathSciNet  Google Scholar 

  7. Davie, A. M., Analytic capacity and approximation problems.Trans. Amer. Math. Soc., 171 (1972), 409–444.

    Article  MATH  MathSciNet  Google Scholar 

  8. Davie, A. M. &Øksendal, B., Analytic capacity and differentiability properties of finely harmonic functions.Acta Math., 149 (1982), 127–152.

    MathSciNet  Google Scholar 

  9. Garnett, J.,Analytic Capacity and Measure. Lecture Notes in Math., 297. Springer-Verlag, Berlin-New York, 1972.

    Google Scholar 

  10. Jones, P. W., Rectifiable sets and the traveling salesman problem.Invent. Math., 102, (1990), 1–15.

    Article  MATH  MathSciNet  Google Scholar 

  11. Jones, P. W. &Murai, T., Positive analytic capacity but zero Buffon needle probability.Pacific J. Math., 133 (1988), 99–114.

    MathSciNet  Google Scholar 

  12. Léger, J. C., Menger curvature and rectifiability.Ann. of Math. (2), 149 (1999), 831–869.

    Article  MATH  MathSciNet  Google Scholar 

  13. Mattila, P., Smooth maps, null-sets for integralgeometric measure and analytic capacity.Ann. of Math. (2), 123 (1986), 303–309.

    Article  MATH  MathSciNet  Google Scholar 

  14. —Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability. Cambridge Stud. Adv. Math., 44, Cambridge Univ. Press, Cambridge, 1995.

    Google Scholar 

  15. Mattila, P. Rectifiability, analytic capacity, and singular integrals, inProceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998).Doc. Math., 1998, Extra Vol. II, 657–664 (electronic).

  16. Mattila, P., Melnikov, M. S. &Verdera, J., The Cauchy integral, analytic capacity, and uniform rectifiability.Ann. of Math. (2), 144 (1996), 127–136.

    Article  MathSciNet  Google Scholar 

  17. Mattila, P. &Paramonov, P. V., On geometric properties of harmonic Lip1-capacity.Pacific J. Math., 171 (1995), 469–491.

    MathSciNet  Google Scholar 

  18. — On density properties of the Riesz capacities and the analytic capacity γ+.Tr. Mat. Inst. Steklova, 235 (2001), 143–156 (Russian); English translation inProc. Steklov Inst. Math., 235 (2001), 136–149.

    MathSciNet  Google Scholar 

  19. Melnikov, M. S., Estimate of the Cauchy integral over an analytic curve.Mat. Sb. (N.S.), 71 (113), (1966), 503–514 (Russian); English translation inAmer. Math. Soc. Transl. Ser. 2, 80 (1969), 243–256.

    MATH  MathSciNet  Google Scholar 

  20. — Analytic capacity: a discrete approach and the curvature of measure.Mat. Sb., 186:6 (1995), 57–76 (Russian); English translation inSb. Math., 186 (1995), 827–846.

    MATH  MathSciNet  Google Scholar 

  21. Melnikov, M. S. & Verdera, J.. A geometric proof of theL 2 boundedness of the Cauchy integral on Lipschitz graphs.Internat. Math. Res. Notices 1995, 325–331.

  22. Mateu, J., Tolsa, X. &Verdera, J., The planar Cantor sets of zero analytic capacity and the localT(b)-theorem.J. Amer. Math. Soc., 16 (2003), 19–28.

    Article  MathSciNet  Google Scholar 

  23. Murai, T.,A Real Variable Method for the Cauchy Transform, and Analytic Capacity. Lecture Notes in Math. 1307. Springer-Verlag, Berlin, 1988.

    Google Scholar 

  24. Nazarov, F., Treil, S. &Volberg, A., TheTb-theorem on non-homogeneous spaces that proves a conjecture of Vitushkin. Preprint, Centre de Recerca Matemàtica, Barcelona, 2002.

    Google Scholar 

  25. — Accretive systemTb-theorems on nonhomogeneous spaces.Duke Math. J., 113 (2002), 259–312.

    Article  MathSciNet  Google Scholar 

  26. Nazarov, F., Treil, S. & Volberg, A. TheTb-theorem on non-homogeneous spaces. To appear inActa Math. (2003).

  27. Pajot, H.,Analytic Capacity, Rectifiability, Menger Curvature and the Cauchy Integral. Lecture Notes in Math., 1799. Springer-Verlag, Berlin, 2002.

    Google Scholar 

  28. Suita, N., On subadditivity of analytic capacity for two continua.Kodai Math. J., 7 (1984), 73–75.

    MATH  MathSciNet  Google Scholar 

  29. Tolsa, X., Curvature of measures, Cauchy singular integral and analytic capacity. Ph.D. Thesis, Universitat Autònoma de Barcelona, 1998.

  30. —L 2-boundedness of the Cauchy integral operator for continuous measures.Duke Math. J., 98 (1999), 269–304.

    Article  MATH  MathSciNet  Google Scholar 

  31. — Principal values for the Cauchy integral and rectifiability.Proc. Amer. Math. Soc., 128 (2000), 2111–2119.

    Article  MATH  MathSciNet  Google Scholar 

  32. — On the analytic capacity γ+.Indiana Univ. Math. J., 51 (2002), 317–343.

    Article  MATH  MathSciNet  Google Scholar 

  33. Verdera, J., Removability, capacity and approximation, inComplex Potential Theory (Montreal, PQ 1993), pp. 419–473. NATO Adv. Sci. Int. Ser. C Math. Phys. Sci., 439. Kluwer, Dordrecht, 1994.

    Google Scholar 

  34. — On theT(1)-theorem for the Cauchy integral.Ark. Mat., 38, (2000), 183–199.

    Article  MATH  MathSciNet  Google Scholar 

  35. Verdera, J., Melnikov, M. S. &Paramonov, P. V.,C 1-approximation and the extension of subharmonic functions.Mat. Sb., 192:4 (2001), 37–58 (Russian); English translation inSb. Math., 192 (2001), 515–535.

    MathSciNet  Google Scholar 

  36. Vitushkin, A. G., The analytic capacity of sets in problems of approximation theory.Uspekhi Mat. Nauk, 22:6 (1967), 141–199. (Russian); English translation inRussian Math. Surveys, 22 (1967), 139–200.

    MATH  Google Scholar 

  37. Vitushkin, A. G. &Melnikov, M. S., Analytic capacity and rational approximation, inLinear and Complex Analysis Problem Book pp. 495–497. Lecture Notes in Math., 1403. Springer-Verlag, Berlin, 1984.

    Google Scholar 

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Authors and Affiliations

  1. Departament de Matemàtiques, Universitat Autònoma de Barcelona, ES-08193 Bellaterra (Barcelona), Bellaterra, Spain

    Xavier Tolsa

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  1. Xavier Tolsa
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Additional information

Supported by a Marie Curie Fellowship of the European Community program Human Potential under contract HPMFCT-2000-00519. Also partially supported by grants DGICYT BFM2000-0361 (Spain) and 2001-SGR-00431 (Generalitat de Catalunya).

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Tolsa, X. Painlevé's problem and the semiadditivity of analytic capacity. Acta Math. 190, 105–149 (2003). https://doi.org/10.1007/BF02393237

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  • DOI: https://doi.org/10.1007/BF02393237

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