Duke Mathematical Journal

$L^2$-boundedness of the Cauchy integral operator for continuous measures

Xavier Tolsa

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Article information

Duke Math. J. Volume 98, Number 2 (1999), 269-304.

First available in Project Euclid: 19 February 2004

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 31A10: Integral representations, integral operators, integral equations methods
Secondary: 30E20: Integration, integrals of Cauchy type, integral representations of analytic functions [See also 45Exx] 31A15: Potentials and capacity, harmonic measure, extremal length [See also 30C85] 47G10: Integral operators [See also 45P05]


Tolsa, Xavier. L 2 -boundedness of the Cauchy integral operator for continuous measures. Duke Math. J. 98 (1999), no. 2, 269--304. doi:10.1215/S0012-7094-99-09808-3. http://projecteuclid.org/euclid.dmj/1077228214.

Export citation


  • [Ah] Lars V. Ahlfors, Bounded analytic functions, Duke Math. J. 14 (1947), 1–11.
  • [Ca] A.-P. Calderón, Cauchy integrals on Lipschitz curves and related operators, Proc. Nat. Acad. Sci. U.S.A. 74 (1977), no. 4, 1324–1327.
  • [Ch] Michael Christ, Lectures on singular integral operators, CBMS Regional Conference Series in Mathematics, vol. 77, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1990, Amer. Math. Soc., Providence.
  • [CMM] R. R. Coifman, A. McIntosh, and Y. Meyer, L'intégrale de Cauchy définit un opérateur borné sur $L\sp2$ pour les courbes lipschitziennes, Ann. of Math. (2) 116 (1982), no. 2, 361–387.
  • [D1] Guy David, Opérateurs intégraux singuliers sur certaines courbes du plan complexe, Ann. Sci. École Norm. Sup. (4) 17 (1984), no. 1, 157–189.
  • [D2] Guy David, Wavelets and singular integrals on curves and surfaces, Lecture Notes in Mathematics, vol. 1465, Springer-Verlag, Berlin, 1991.
  • [DS1] G. David and S. Semmes, Singular integrals and rectifiable sets in $\bf R\sp n$: Beyond Lipschitz graphs, Astérisque (1991), no. 193, 152, Soc. Math. France, Montrouge.
  • [DS2] Guy David and Stephen Semmes, Analysis of and on uniformly rectifiable sets, Mathematical Surveys and Monographs, vol. 38, American Mathematical Society, Providence, RI, 1993.
  • [Ga] John Garnett, Analytic capacity and measure, Lecture Notes in Math., vol. 297, Springer-Verlag, Berlin, 1972.
  • [Gu] Miguel de Guzmán, Real variable methods in Fourier analysis, North-Holland Mathematics Studies, vol. 46, North-Holland Publishing Co., Amsterdam, 1981.
  • [L] J.-C. Léger, Courbure de Menger et rectifiabilité, Ph.D. thesis, Université de Paris-Sud, Orsay, 1997.
  • [Ma1] Pertti Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995.
  • [Ma2] Pertti Mattila, On the analytic capacity and curvature of some Cantor sets with non-$\sigma$-finite length, Publ. Mat. 40 (1996), no. 1, 195–204.
  • [MMV] Pertti Mattila, Mark S. Melnikov, and Joan Verdera, The Cauchy integral, analytic capacity, and uniform rectifiability, Ann. of Math. (2) 144 (1996), no. 1, 127–136.
  • [Me1] M. S. Melnikov, Estimate of the Cauchy integral over an analytic curve, Mat. Sb. (N.S.) 71 (113) (1966), 503–514, (in Russian); English transl. in Amer. Math. Soc. Transl. Ser. 2 80 (1969), 243–256.
  • [Me2] M. S. Melnikov, Analytic capacity: a discrete approach and the curvature of measure, Mat. Sb. 186 (1995), no. 6, 57–76, (in Russian); English transl. in Russian Acad. Sci. Sb. Math. 186 (1995), 827–846.
  • [MV] Mark S. Melnikov and Joan Verdera, A geometric proof of the $L\sp 2$ boundedness of the Cauchy integral on Lipschitz graphs, Internat. Math. Res. Notices (1995), no. 7, 325–331.
  • [Mu] Takafumi Murai, A real variable method for the Cauchy transform, and analytic capacity, Lecture Notes in Mathematics, vol. 1307, Springer-Verlag, Berlin, 1988.
  • [NTV1] F. Nazarov, S. Treil, and A. Volberg, Boundedness of the Cauchy integral, preprint, 1997.
  • [NTV2] F. Nazarov, S. Treil, and A. Volberg, Pulling ourselves up by the hair, preprint, 1997.
  • [NTV3] F. Nazarov, S. Treil, and A. Volberg, personal communication.
  • [S1] Eric Sawyer, A two weight weak type inequality for fractional integrals, Trans. Amer. Math. Soc. 281 (1984), no. 1, 339–345.
  • [S2] Eric T. Sawyer, A characterization of two weight norm inequalities for fractional and Poisson integrals, Trans. Amer. Math. Soc. 308 (1988), no. 2, 533–545.
  • [SW] E. Sawyer and R. L. Wheeden, Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces, Amer. J. Math. 114 (1992), no. 4, 813–874.
  • [St1] Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970.
  • [St2] Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993.
  • [StW] Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, vol. 32, Princeton University Press, Princeton, N.J., 1971.
  • [Su] Nobuyuki Suita, On subadditivity of analytic capacity for two continua, Kodai Math. J. 7 (1984), no. 1, 73–75.
  • [Ve1] Joan Verdera, A weak type inequality for Cauchy transforms of finite measures, Publ. Mat. 36 (1992), no. 2B, 1029–1034 (1993).
  • [Ve2] J. Verdera, A weak type inequality for Cauchy transforms of measures, preprint, 1992.
  • [Vi] A. G. Vituškin, Analytic capacity of sets in problems of approximation theory, Uspehi Mat. Nauk 22 (1967), no. 6 (138), 141–199, (in Russian); English transl. in Russian Math. Surveys 22, no. 6 (1967), 139–200.
  • [VM] A. G. Vitushkin and M. S. Melnikov, “Analytic capacity and rational approximation“, Linear and Complex Analysis Problem Book: 199 Research Problems, Lecture Notes in Math., vol. 1043, Springer-Verlag, Berlin, 1984, pp. 495–497.