On boundary behavior of Cauchy integrals

Hiroshige Shiga

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica (2013)

  • Volume: 67, Issue: 1
  • ISSN: 0365-1029

Abstract

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In this paper, we shall estimate the growth order of the n-th derivative Cauchy integrals at a point in terms of the distance between the point and the boundary of the domain. By using the estimate, we shall generalize Plemelj–Sokthoski theorem. We also consider the boundary behavior of generalized Cauchy integrals on compact bordered Riemann surfaces.

How to cite

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Hiroshige Shiga. "On boundary behavior of Cauchy integrals." Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica 67.1 (2013): null. <http://eudml.org/doc/289720>.

@article{HiroshigeShiga2013,
abstract = {In this paper, we shall estimate the growth order of the n-th derivative Cauchy integrals at a point in terms of the distance between the point and the boundary of the domain. By using the estimate, we shall generalize Plemelj–Sokthoski theorem. We also consider the boundary behavior of generalized Cauchy integrals on compact bordered Riemann surfaces.},
author = {Hiroshige Shiga},
journal = {Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica},
keywords = {Cauchy integral; Plemelj-Sokthoski theorem; Riemann surface.},
language = {eng},
number = {1},
pages = {null},
title = {On boundary behavior of Cauchy integrals},
url = {http://eudml.org/doc/289720},
volume = {67},
year = {2013},
}

TY - JOUR
AU - Hiroshige Shiga
TI - On boundary behavior of Cauchy integrals
JO - Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
PY - 2013
VL - 67
IS - 1
SP - null
AB - In this paper, we shall estimate the growth order of the n-th derivative Cauchy integrals at a point in terms of the distance between the point and the boundary of the domain. By using the estimate, we shall generalize Plemelj–Sokthoski theorem. We also consider the boundary behavior of generalized Cauchy integrals on compact bordered Riemann surfaces.
LA - eng
KW - Cauchy integral; Plemelj-Sokthoski theorem; Riemann surface.
UR - http://eudml.org/doc/289720
ER -

References

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  2. Bikcantaev, I. A., Analogues of a Cauchy kernel on a Riemann surface and some applications of them, Mat. Sb. (N.S.) 112 (154), no. 2 (6) (1980), 256–282 (Russian); translation in Math. USSR Sb. 40, no. 2 (1981), 241–265. 
  3. Block, I. E., The Plemelj theory for the class of functions, Duke Math. J. 19 (1952), 367–378. 
  4. Duren, P. L., Theory of Hp Spaces, Academic Press, New York–San Francisco–London, 1970. 
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  6. Gakhov, F. D., Boundary Value Problems, Pergamon Press, Oxford–New York–Paris, 1966. 
  7. Garnett, J. B., Bounded Analytic Functions, Academic Press, New York–London, 1981. 
  8. Gong, S., Integrals of Cauchy type on the ball, International Press, Cambridge, 1993. 
  9. Guseinov, E. G., The Plemelj-Privalov theorem for generalized Holder classes, Mat. Sb. 183, no. 2 (1992), 21–37 (Russian); translation in Russian Acad. Sci. Sb. Math. 75 (1993), 165–182. 
  10. Heins, M., Hardy Classes on Riemann Surfaces, Springer-Verlag, Berlin–New York, 1969. 
  11. Shiga, H., Riemann mappings of invariant components of Kleinian groups, J. London Math. Soc. 80 (2009), 716–728. 
  12. Shiga, H., Modulus of continuity, a Hardy–Littlewood theorem and its application, RIMS Kokyuroku Bessatsu, 2010, 127–133. 
  13. Pommerenke, C., Boundary Behaviour of Conformal Maps, Springer-Verlag, Berlin, 1992. 
  14. Walsh, J. L., Polynomial expansions of functions defined by Cauchy’s integral, J. Math. Pures Appl. 31 (1952), 221–244. 

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