Painlevé's problem and the semiadditivity of analytic capacity

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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).

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

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

— 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.

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.

— 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.

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.

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.

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

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.

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

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

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

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

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

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

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

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

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.

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

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.

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.

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.