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Principal Values of Cauchy Integrals, Rectifiable Measures and Sets

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ICM-90 Satellite Conference Proceedings

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Abstract

The extensive studies started by A. P. Calderon in the sixties and continued by many authors up today have revealed that the Cauchy integrals

$$ {C_{\Gamma }}f(z) = \int_{\Gamma } {\frac{{f\left( \zeta \right)d\zeta }}{{\zeta - z}}} $$

behave very well on sufficiently regular, not necessarily smooth, curves F, see [CCFJR], [D] and [MT].

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References

  1. A. P. Calderon, C. P. Calderon, E. B. Fabes, M. Jodeit Jr. and N. M. Riviere, Applications of the Cauchy integrals along Lipschitz curves, Bull. Amer. Math. Soc. 84 (1978), 287–290.

    Article  MathSciNet  MATH  Google Scholar 

  2. G. David, Operateurs integraux singuliers sur certaines courbes du plan, Ann. Sci. Ecole Norm. Sup. (4) 17 (1984), 157–189.

    MathSciNet  MATH  Google Scholar 

  3. K. J. Falconer, Geometry of Fractal Sets, Cambridge University Press, 1985.

    Google Scholar 

  4. X. Fang, The Cauchy integral of Calderon and analytic capacity, Ph. D. Thesis, Yale University, 1990.

    Google Scholar 

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

    Google Scholar 

  6. P. W. Jones, Square functions, Cauchy integrals, analytic capacity, and harmonic measure, (Harmonic Analysis and Partial Differential Equations (ed. J. Garcia-Cuerva), Lecture Notes in Math. 1384, Springer-Verlag, 1989, pp. 24–68.

    Google Scholar 

  7. D. Khavinson, On a geometric localization of the Cauchy potentials, Michigan Math. J. 33 (1986), 377–385.

    MathSciNet  MATH  Google Scholar 

  8. P. Mattila, A class of sets with zero length and positive analytic capacity, Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), 387–395.

    MathSciNet  MATH  Google Scholar 

  9. MP2] P. Mattila, Cauchy singular integrals and rectifiability of measures in the plane, to appear in Adv. in Math.

    Google Scholar 

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

    Google Scholar 

  11. D. Preiss, Geometry of measures in Rn. Distribution, rectifiability, and densities, Ann. of Math. 125 (1987), 537–643.

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Jyväskylä, P.O. Box 35, SF-40351, Jyväskylä, Finland

    Pertti Mattila

Authors
  1. Pertti Mattila
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Editor information

Editors and Affiliations

  1. Mathematical Institute, Tohoku University, Aobaku, Sendai, Miyagi, 980, Japan

    Satoru Igari

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© 1991 Springer-Verlag Tokyo

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Mattila, P. (1991). Principal Values of Cauchy Integrals, Rectifiable Measures and Sets. In: Igari, S. (eds) ICM-90 Satellite Conference Proceedings. Springer, Tokyo. https://doi.org/10.1007/978-4-431-68168-7_14

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  • DOI: https://doi.org/10.1007/978-4-431-68168-7_14

  • Publisher Name: Springer, Tokyo

  • Print ISBN: 978-4-431-70084-5

  • Online ISBN: 978-4-431-68168-7

  • eBook Packages: Springer Book Archive

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