L2-boundedness of the cauchy integral operator for continuous measures

Duke Mathematical Journal

Volumn 98, Issue 2, 1999, Pages 269-304

  Tolsa, Xavier ^a  

^a UNIVERSITY OF BARCELONA   (Spain)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0033147832     PISSN: 00127094     EISSN: None     Source Type: Journal    
DOI: 10.1215/S0012-7094-99-09808-3     Document Type: Article

References (35)

1. L. Ahlfors, Bounded analytic functions, Duke Math. J. 14 (1947), 1-11.
2. A. P. Calderón, Cauchy integrals on Lipschitz curves and related operators, Proc. Nat. Acad. Sci. U.S.A. 74 (1977), 1324-1327.
3. M. Christ, Lectures on Singular Integral Operators, CBMS Regional Conf. Ser. in Math. 77, Amer. Math. Soc., Providence, 1990.
4. 2 pour les courbes lipschitziennes, Ann. of Math. (2) 116 (1982), 361-387.
5. G. David, Opérateurs intégraux singuliers sur certaines courbes du plan complexe, Ann. Sci. École Norm. Sup. (4) 17 (1984), 157-189.
6. G. David, Wavelets and Singular Integrals on Curves and Surfaces, Lecture Notes in Math. 1465, Springer-Verlag, Berlin, 1991.
7. G. David and S. Semmes, Singular Integrals and Rectifiable Sets in R": Beyond Lipschitz Graphs, Astérisque 193, Soc. Math. France, Montrouge, 1991.
8. G. David and S. Semmes, Analysis of and on Uniformly Rectifiable Sets, Math. Surveys Monogr. 38, Amer. Math. Soc., Providence, 1993.
9. J. Garnett, Analytic Capacity and Measure, Lecture Notes in Math. 297, Springer-Verlag, Berlin, 1972.
10. M. de Guzmán, Real Variable Methods in Fourier Analysis, North-Holland Math. Stud. 46, North-Holland, Amsterdam, 1981.
11. J.-C. Léger, Courbure de Menger et rectifiabilité, Ph.D. thesis, Université de Paris-Sud, Orsay, 1997.
12. P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Cambridge Stud. Adv. Math. 44, Cambridge Univ. Press, Cambridge, 1995.
13. P. Mattila, On the analytic capacity and curvature of some Cantor sets with non-o –finite length, Publ. Mat. 40 (1996), 195-204.
14. P. Mattila, M. S. Melnikov, and J. Verdera, The Cauchy integral, analytic capacity, and uniform rectifiability, Ann. of Math. (2) 144 (1996), 127-136.
15. M. S. Melnikov, Estimate of the Cauchy integral over an analytic curve (in Russian), Mat. Sb. (N.S.) 71 (1966), 503-514;
16. M. S. Melnikov, English transl. in Amer. Math. Soc. Transl. Ser. 2 80 (1969), 243-256.
17. M. S. Melnikov, Analytic capacity: discrete approach and curvature of measure (in Russian), Mat.Sb. 186 (1995), 57-76;
18. M. S. Melnikov, English transl. in Russian Acad. Sci. Sb. Math. 186 (1995), 827-846.
19. 2 boundedness of the Cauchy integral on Lipschitz graphs, Internat. Math. Res. Notices 1995, 325-331.
20. T. Murai, A Real Variable Method for the Cauchy Transform, and Analytic Capacity, Lecture Notes in Math. 1307, Springer-Verlag, Berlin, 1988.
21. F. Nazarov, S. Treil, and A. Volberg, Boundedness of the Cauchy integral, preprint, 1997.
22. F. Nazarov, S. Treil, and A. Volberg, Pulling ourselves up by the hair, preprint, 1997.
23. F. Nazarov, S. Treil, and A. Volberg, personal communication.
24. E. Sawyer, A two weight weak type inequality for fractional integrals, Trans. Amer. Math. Soc. 281 (1984), 339-345.
25. E. Sawyer, A characterization of two weight norm inequalities for fractional and Poisson integrals, Trans. Amer. Math. Soc. 308 (1988), 533-545.
26. E. Sawyer and R. L. Wheeden, Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces, Amer. J. Math. 114 (1992), 813-874.
27. E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Math. Ser. 30, Princeton Univ. Press, Princeton, 1970.
28. E. Sawyer, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Math. Ser. 43, Princeton Univ. Press, Princeton, 1993.
29. E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Math. Ser. 32, Princeton Univ. Press, Princeton, 1971.
30. N. Suita, On subadditivity of analytic capacity for two continua, Kodai Math. J. 7 (1984), 73- 75.
31. J. Verdera, A weak type inequality for Cauchy transforms of finite measures, Publ. Mat. 36 (1992), 1029-1034.
32. J. Verdera, A weak type inequality for Cauchy transforms of measures, preprint, 1992.
33. A. G. Vitushkin, The analytic capacity of sets in problems of approximation theory (in Russian), Uspekhi Mat. Nauk 22, no. 6 (1967), 141-199;
34. A. G. Vitushkin, English transl. in Russian Math. Surveys 22, no. 6 (1967), 139-200.
35. A. G. Vitushkin and M. S. Melnikov, “Analytic capacity and rational approximation” in Linear and Complex Analysis Problem Book: 199 Research Problems, Lecture Notes in Math. 1043, Springer-Verlag, Berlin, 1984, 495-497.