Geodesic
Saga of the Painlevé Problem and Analytic Capacity
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analytic and geometric issues of complex analysis, Tome 235 (2001), pp. 157-164
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This note consists of two sections. The first one gives an account of an intriguing and dramatic story of solving (not completely) the so-called Painlevé problem that consists in describing the set of removable singularities for bounded holomorphic functions. In view of this, I indulge in proposing some reminiscences about bygone events. The second section gives yet another elementary proof of the Denjoy conjecture, which is a part of the Painlevé problem. 
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M. S. Mel'nikov. Saga of the Painlevé Problem and Analytic Capacity. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analytic and geometric issues of complex analysis, Tome 235 (2001), pp. 157-164. http://geodesic.mathdoc.fr/item/TM_2001_235_a10/

[1] Painlevé P., “Sur les lignes singulières des fonctions analytiques”, Ann. Fac. Sci. Toulouse, 2 (1888) | MR | Zbl

[2] Denjoy A., “Sur les fonctions analytiques uniformes á singularities discontinues”, C. R. Acad. Sci. Paris, 149 (1909), 258–260

[3] Ahlfors L. V., Beurling A., “Conformal invariants and function theoretic null sets”, Acta Math., 83 (1950), 101–129 | DOI | MR | Zbl

[4] Ivanov L. D., “O gipoteze Danzhua”, UMN, 18:4 (1963), 147–149 | MR | Zbl

[5] Davie A. M., “Analytic capacity and approximation problems”, Trans. Amer. Math. Soc., 171 (1972), 409–444 | DOI | MR | Zbl

[6] Besicovitch A., “On the fundamental geometric properties of linearly mesurable plane sets of points, I, II”, Math. Ann., 98 (1927), 422–464 ; 115 (1938), 296–329 | DOI | MR | Zbl | DOI | MR | Zbl

[7] Besicovitch A., “On the fundamental geometric properties of linearly mesurable plane sets of points, III”, Math. Ann., 116 (1939), 349–357 | DOI | MR | Zbl

[8] Ahlfors L. V., “Bounded analytic functions”, Duke Math. J., 14 (1947), 1–11 | DOI | MR | Zbl

[9] Garabedian P. R., “Schwarz's lemma and the Szegö kernel function”, Trans. Amer. Math. Soc., 67 (1949), 1–35 | DOI | MR | Zbl

[10] Vitushkin A. G., “Analiticheskaya emkost mnozhestv v zadachakh teorii priblizhenii”, UMN, 22 (1967), 141–199

[11] Vitushkin A. G., “Primer mnozhestva polozhitelnoi dliny, no nulevoi analiticheskoi emkosti”, DAN SSSR, 127 (1959), 246–249 | Zbl

[12] Ivanov L. D., Variatsii mnozhestv i funktsii, Nauka, M., 1975 | MR

[13] Garnett J., “Positive length but zero analytic capacity”, Proc. Amer. Math. Soc., 24 (1970), 696–699 | DOI | MR | Zbl

[14] Calderon A. P., “Cauchy integrals on Lipschitz curves and related operators”, Proc. Nat. Acad. Sci. USA, 74 (1977), 1324–1327 | DOI | MR | Zbl

[15] Khavin V. P., “Granichnye svoistva integralov tipa Koshi i garmonicheski sopryazhennykh funktsii v oblastyakh so spryamlyaemoi granitsei”, Mat. sb., 68 (1965), 499–517 | Zbl

[16] Khavin V. P., Khavinson S. Ya., “Nekotorye otsenki analiticheskoi emkosti”, DAN SSSR, 138 (1961), 789–792 | Zbl

[17] Mattila P., “Smooth maps, null-sets for integral geometric measure and analytic capacity”, Ann. Math., 123 (1986), 303–309 | DOI | MR | Zbl

[18] Jones P. W., Murai T., “Positive analytic capacity but zero Buffon needle probability”, Pasif. J. Math., 133 (1988), 99–114 | MR | Zbl

[19] Melnikov M. S., “Analiticheskaya emkost: diskretnyi podkhod i krivizna mery”, Mat. sb., 186:6 (1995), 57–76 | MR

[20] Melnikov M. S., Verdera J., “A geometric proof of $L^2$ boundedness of the Cauchy integral on Lipschitz curves”, Intern. Math. Res. Not., 7 (1995), 325–331 | DOI | MR | Zbl

[21] Mattila P., Melnikov M. S., Verdera J., “The Cauchy integral, analytic capacity and uniform rectifiability”, Ann. Math., 144 (1996), 127–136 | DOI | MR | Zbl

[22] Christ M., “$T(b)$ theorem with remarks on analytic capacity and the Cauchy integral”, Colloq. Math., 60/61 (1990), 601–628 | MR | Zbl

[23] David G., Semmes S., Singular integrals and rectifiable sets in $\mathbb{R}^n$, Astérisque, 193, Acad. Publ., Paris, 1991 | MR

[24] Léger J.-C., “Menger curvature and rectifiability”, Ann. Math., 149 (1999), 831–869 | DOI | MR | Zbl

[25] David G., Mattila P., “Removable sets for Lipschitz harmonic functions in the plane”, Rev. Mat. Iberoamer, 16 (2000), 137–215 | MR | Zbl

[26] David G., “Unrectifiable $1$-sets have vanishing analytic capacity”, Rev. Mat. Iberoamer, 14 (1998), 369–479 | MR | Zbl

[27] Nazarov F., Treil S., Volberg A., Pulling ourselves by the hair (the proof of the Vitushkin conjecture), Preprint, Michigan State Univ., 2000

[28] Tolsa X., “$L^2$-boundedness of the Cauchy integral operators for continuous measures”, Duke Math. J., 98:2 (1999), 269–304 | DOI | MR | Zbl