Geodesic
On boundary behavior of Cauchy integrals
Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 67 (2013) no. 1.

Voir la notice de l'article provenant de la source Library of Science

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.
Keywords: Cauchy integral, Plemelj-Sokthoski theorem, Riemann surface. 
@article{AUM_2013_67_1_a3,
     author = {Shiga, Hiroshige},
     title = {On boundary behavior of {Cauchy} integrals},
     journal = {Annales Universitatis Mariae Curie-Sk{\l}odowska. Mathematica },
     publisher = {mathdoc},
     volume = {67},
     number = {1},
     year = {2013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/AUM_2013_67_1_a3/}
}
TY  - JOUR
AU  - Shiga, Hiroshige
TI  - On boundary behavior of Cauchy integrals
JO  - Annales Universitatis Mariae Curie-Skłodowska. Mathematica 
PY  - 2013
VL  - 67
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/AUM_2013_67_1_a3/
LA  - en
ID  - AUM_2013_67_1_a3
ER  - 
%0 Journal Article
%A Shiga, Hiroshige
%T On boundary behavior of Cauchy integrals
%J Annales Universitatis Mariae Curie-Skłodowska. Mathematica 
%D 2013
%V 67
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/AUM_2013_67_1_a3/
%G en
%F AUM_2013_67_1_a3
Shiga, Hiroshige. On boundary behavior of Cauchy integrals. Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 67 (2013) no. 1. http://geodesic.mathdoc.fr/item/AUM_2013_67_1_a3/

[1] Aikawa, H., Modulus of continuity of the Dirichlet solutions, Bull. London Math. Soc. 42 (2010), 857–867.

[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.

[5] Farkas, H. M., Kra, I., Riemann Surfaces, Springer-Verlag, New York–Heidelberg–Berlin, 1980.

[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.