Analytic capacity and the Painlevé Problem
Séminaire Bourbaki (2003-2004)
- Volume: 46, page 301-328
- ISSN: 0303-1179
Access Full Article
topAbstract
topHow to cite
topPajot, Hervé. "Capacité analytique et le problème de Painlevé." Séminaire Bourbaki 46 (2003-2004): 301-328. <http://eudml.org/doc/252170>.
@article{Pajot2003-2004,
abstract = {Le problème de Painlevé consiste à trouver une caractérisation géométrique des sous-ensembles du plan complexe qui sont effaçables pour les fonctions holomorphes bornées. Ce problème d’analyse complexe a connu ces dernières années des avancées étonnantes, essentiellement grâce au dévelopement de techniques fines d’analyse réelle et de théorie de la mesure géométrique. Dans cet exposé, nous allons présenter et discuter une solution proposée par X. Tolsa en termes de courbure de Menger au problème de Painlevé.},
author = {Pajot, Hervé},
journal = {Séminaire Bourbaki},
keywords = {analytic capacity; rectifiability; Cauchy integral; Menger curvature; uniformly rectifiable sets},
language = {fre},
pages = {301-328},
publisher = {Association des amis de Nicolas Bourbaki, Société mathématique de France},
title = {Capacité analytique et le problème de Painlevé},
url = {http://eudml.org/doc/252170},
volume = {46},
year = {2003-2004},
}
TY - JOUR
AU - Pajot, Hervé
TI - Capacité analytique et le problème de Painlevé
JO - Séminaire Bourbaki
PY - 2003-2004
PB - Association des amis de Nicolas Bourbaki, Société mathématique de France
VL - 46
SP - 301
EP - 328
AB - Le problème de Painlevé consiste à trouver une caractérisation géométrique des sous-ensembles du plan complexe qui sont effaçables pour les fonctions holomorphes bornées. Ce problème d’analyse complexe a connu ces dernières années des avancées étonnantes, essentiellement grâce au dévelopement de techniques fines d’analyse réelle et de théorie de la mesure géométrique. Dans cet exposé, nous allons présenter et discuter une solution proposée par X. Tolsa en termes de courbure de Menger au problème de Painlevé.
LA - fre
KW - analytic capacity; rectifiability; Cauchy integral; Menger curvature; uniformly rectifiable sets
UR - http://eudml.org/doc/252170
ER -
References
top- [1] L. Ahlfors – “Bounded analytic functions”, Duke Math. J.14 (1947), p. 1–11. Zbl0030.03001MR21108
- [2] L. Ahlfors & A. Beurling – “Conformal invariants and function theoretic null-sets”, Acta Math.83 (1950), p. 101–129. Zbl0041.20301MR36841
- [3] A.P. Calderón – “Cauchy integrals on Lipschitz curves and related operators”, Proc. Nat. Acad. Sci. U.S.A.74 (1977), p. 1324–1327. Zbl0373.44003MR466568
- [4] M. Christ – “A theorem with remarks on analytic capacity and the Cauchy integral”, Colloq. Math. 60-61 (1990), p. 601–628. Zbl0758.42009MR1096400
- [5] —, Lectures on singular integral operators, Regional Conference Series in Mathematics, vol. 77, American Mathematical Society, 1990. Zbl0745.42008MR1104656
- [6] R. Coifman, A. McIntosh & Y. Meyer – “L’opérateur de Cauchy définit un opérateur borné sur sur les courbes lipschitziennes”, Ann. of Math.116 (1982), p. 361–388. Zbl0497.42012MR672839
- [7] R. Coifman & G. Weiss – Analyse harmonique non-commutative sur certains espaces homogènes, Lect. Notes in Math., vol. 242, Springer-Verlag, 1971. Zbl0224.43006MR499948
- [8] G. David – Wavelets and singular integral operators on curves and surfaces, Lect. Notes in Math., vol. 1465, Springer-Verlag, 1991. Zbl0764.42019MR1123480
- [9] —, “Unrectifiable -sets have vanishing analytic capacity”, Rev. Mat. Iberoamericana14 (1998), p. 369–479. Zbl0913.30012MR1654535
- [10] G. David & P. Mattila – “Removable sets for Lipschitz harmonic functions in the plane”, Rev. Mat. Iberoamericana16 (2000), p. 137–215. Zbl0976.30016MR1768535
- [11] G. David & S. Semmes – Singular integrals and rectifiable sets in : Au-delà des graphes lipschitziens, Astérisque, vol. 193, Société Mathématique de France, 1991. Zbl0743.49018
- [12] —, Analysis of and on uniformly rectifiable sets, Mathematical Surveys and Monographs, vol. 38, American Mathematical Society, 1993. Zbl0832.42008MR1251061
- [13] —, “Quantitative rectifiability and Lipschitz mappings”, Trans. Amer. Math. Soc.337 (1993), p. 855–889. Zbl0792.49029MR1132876
- [14] A. Denjoy – “Sur les fonctions analytiques uniformes à singularités discontinues”, C.R. Acad. Sci. Paris149 (1909), p. 258–260. Zbl40.0442.05JFM40.0442.05
- [15] H. Farag – “The Riesz kernels do not give rise to higher dimensional analogues to the Menger-Melnikov curvature”, Publ. Mat.43 (1999), p. 251–260. Zbl0936.42010MR1697524
- [16] J.B. Garnett – “Positive length but zero analytic capacity”, Proc. Amer. Math. Soc.24 (1970), p. 696–699. Zbl0208.09803
- [17] J.B. Garnett & J. Verdera – “Analytic capacity, bilipschitz maps and Cantor sets”, Math. Res. Lett.10 (2003), p. 515–522. Zbl1063.30025MR1995790
- [18] P. Jones – “Rectifiable sets and the traveling salesman problem”, Invent. Math.102 (1990), p. 1–15. Zbl0731.30018MR1069238
- [19] P. Jones & T. Murai – “Positive analytic capacity, but zero Buffon needle probability”, Pacific J. Math.133 (1988), p. 99–114. Zbl0653.30016MR936358
- [20] H. Joyce & P. Mörters – “A set with finite curvature and projections of zero length”, J. Math. Anal. Appl.247 (2000), p. 126–135. Zbl0973.30022MR1766928
- [21] J.-C. Léger – “Rectifiability and Menger curvature”, Ann. of Math.149 (1999), p. 831–869. Zbl0966.28003MR1709304
- [22] J. Mateu, L. Prat & J. Verdera – “The capacity associated to signed Riesz kernels and Wolff potentials”, J. reine angew. Math. 578 (2005), p. 201–223. Zbl1086.31005MR2113895
- [23] J. Mateu & X. Tolsa – “Riesz transforms and harmonic -capacity of Cantor sets”, Proc. London Math. Soc. (3) 89 (2004), p. 676–696. Zbl1089.42009MR2107011
- [24] J. Mateu, X. Tolsa & J. Verdera – “The planar Cantor sets of zero analytic capacity and the local theorem”, J. Amer. Math. Soc.16 (2003), p. 19–28. Zbl1016.30020MR1937197
- [25] P. Mattila – “Smooth maps, null-sets for integral geometric measures and analytic capacity”, Ann. of Math.123 (1986), p. 303–309. Zbl0589.28006MR835764
- [26] —, “Orthogonal projections, Riesz capacities, and Minkowski content”, Indiana Univ. Math. J.39 (1990), p. 185–198. Zbl0682.28003MR1052016
- [27] —, Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, 1995. Zbl0819.28004MR1333890
- [28] —, “On the analytic capacity and curvature of some Cantor sets with non--finite length”, Publ. Mat.40 (1996), p. 195–204. Zbl0888.30026MR1397014
- [29] —, “Hausdorff dimension, projections, and the Fourier transform”, Publ. Mat.48 (2004), p. 3–48. Zbl1049.28007MR2044636
- [30] P. Mattila, M. Melnikov & J. Verdera – “The Cauchy integral, analytic capacity, and uniform rectifiability”, Ann. of Math.144 (1996), p. 127–136. Zbl0897.42007MR1405945
- [31] P. Mattila & P.V. Paramonov – “On geometric properties of harmonic -capacity”, Pacific J. Math.171 (1995), p. 469–491. Zbl0852.31004MR1372240
- [32] M. Melnikov – “Analytic capacity : discrete approach and curvature of measure”, Sb. Math.186 (1995), p. 827–846. Zbl0840.30008MR1349014
- [33] M. Melnikov & J. Verdera – “A geometric proof of the boundedness of the Cauchy integral on Lipschitz curves”, Internat. Math. Res. Notices7 (1995), p. 325–331. Zbl0923.42006MR1350687
- [34] F. Nazarov, S. Treil & A. Volberg – “Cauchy integral and Calderón-Zygmund operators on nonhomogeneous spaces”, Internat. Math. Res. Notices15 (1997), p. 703–726. Zbl0889.42013MR1470373
- [35] —, “Nonhomogeneous theorem which proves Vitushkin’s conjecture”, Preprint no 519, CRM Barcelona, 2002.
- [36] —, “ theorems on nonhomogeneous spaces”, Acta Math.190 (2003), p. 151–239.
- [37] K. Okikiolu – “Characterizations of subsets of rectifiable curves in ”, J. London Math. Soc. (2) 46 (1992), p. 336–348. Zbl0758.57020MR1182488
- [38] P. Painlevé – Leçons sur la théorie analytique des équations différentielles professées à Stockholm, Hermann, 1897. JFM28.0262.01
- [39] H. Pajot – “Conditions quantitatives de rectifiabilité”, Bull. Soc. math. France 125 (1997), p. 15–53. Zbl0890.28004MR1459297
- [40] —, Analytic capacity, rectifiability, Menger curvature and the Cauchy integral, Lect. Notes in Math., vol. 1799, Springer-Verlag, Berlin, 2002. Zbl1043.28002MR1952175
- [41] —, “Le problème géométrique du voyageur de commerce, et ses applications à l’analyse complexe et harmonique”, in Autour du centenaire Lebesgue, Panoramas & Synthèses, vol. 18, Société Mathématique de France, 2004, p. 123–156. Zbl1108.30019
- [42] Y. Peres & B. Solomyak – “How likely is Buffon’s needle to fall near a planar Cantor set ?”, Pacific J. Math.204 (2002), p. 473–496. Zbl1046.28006MR1907902
- [43] L. Prat – “Potential theory of signed Riesz kernels : Capacity and Hausdorff measures”, Internat. Math. Res. Notices19 (2004), p. 937–981. Zbl1082.31002MR2037051
- [44] R. Schul – “Subset of rectifiable curves in Hilbert space and the Analyst’s TSP”, Thèse, Yale University, 2005. Zbl1152.28006MR2707104
- [45] E.M. Stein – Harmonic analysis : real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, 1993. Zbl0821.42001MR1232192
- [46] X. Tolsa – “-boundedness of the Cauchy integral for continuous measures”, Duke Math. J.98 (1999), p. 269–304. Zbl0945.30032MR1695200
- [47] —, “On the analytic capacity ”, Indiana Univ. Math. J.51 (2002), p. 317–343. Zbl1041.31002MR1909292
- [48] —, “Painlevé’s problem and the semiadditivity of analytic capacity”, Acta Math.190 (2003), p. 105–149. Zbl1060.30031MR1982794
- [49] —, “The boundedness of the Cauchy transform implies boundedness of all antisymmetric Calderón-Zygmund operators”, Publ. Mat.48 (2004), p. 445–479. Zbl1066.42013MR2091015
- [50] —, “Bilipschitz maps, analytic capacity, and the Cauchy integral”, Ann. of Math. (2) (à paraître). Zbl1097.30020
- [51] J. Verdera – “The Fall of the doubling condition in Calderón-Zygmund theory”, in Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations (El Escorial 2000), Publ. Mat., 2002, numéro spécial, p. 275–292. Zbl1025.42008MR1964824
- [52] A.G. Vitushkin – “The analytic capacity of sets in approximation theory”, Uspekhi Mat. Nauk 22 (1967), p. 141–199, en russe ; traduction en anglais dans Russian Math. Surveys 22 (1967), p. 139-200. Zbl0164.37701MR229838
- [53] A. Volberg – Calderón-Zygmund capacities and operators on nonhomogeneous spaces, Regional Conference Series in Mathematics, vol. 100, American Mathematical Society, 2003. Zbl1053.42022MR2019058
Citations in EuDML Documents
topNotesEmbed ?
topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.