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Cauchy Integral of Calderón on the Graphs of Functions with BMO Derivatives

Published online by Cambridge University Press:  20 November 2018

Kôzô Yabuta
Kôzô Yabuta*
Affiliation:
Department of Mathematics, Ibaraki University, Mito, Ibaraki 310, Japan
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Abstract

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We first note that each graph (x,A(x)) of a function A(x) with BMO derivative is a chord-arc curve. Using this, Muckenhoupt's Ap theory, and the theory of Calderón-Zygmund operators, we shall derive weighted norm inequalities for the Cauchy integral on such graphs from a recent theorem of G. David on the L2-boundedness of Cauchy integral on almost-lipschitzian curves.

Keywords

42B20
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 28 , Issue 4 , 01 December 1985 , pp. 495 - 500
Copyright
Copyright © Canadian Mathematical Society 1985

References

1. Coifman, R.R. and Fefferman, C., Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51, (1974), pp. 241–250.Google Scholar
2. Coifman, R.R. and Meyer, Y., Une généralisation du théorème de Calderón sur l'intégrale de Cauchy, Fourier Analysis (Proc. Sem., El Escorial, 1979), pp. 87–116, Asoc. Mat. Española, Madrid, 1980.Google Scholar
3. Coifman, R.R., David, G. and Meyer, Y., La solution des conjectures de Calderón, Adv. in Math., 48, (1983), pp. 144–148.Google Scholar
4. David, G., Opérateurs intégraux singuliers sur certaines courbes du plan complexe, (preprint).Google Scholar
5. John, F. and Nirenberg, L., On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14, (1961), pp. 415–426.Google Scholar
6. Jones, P., Homeomorphism of the real line which preserves BMO, Ark. Mat. 21, (1983), pp. 229—231.Google Scholar
7. Journé, J-L., Calderón—Zygmund operators, Pseudo-differential operators and the Cauchy integral of Calderón, Lecture Notes in Math., Vol. 994, Springer-Verlag, Berlin Heidelberg, 1983.Google Scholar
8. Krickeles, B.C., Weighted Lp estimates for the Cauchy integral operator, Michigan Math. J. 30, (1983), pp. 231–244.Google Scholar
9. Murai, T., Boundedness of singular integral operators of Calderón type, III, Preprint series No. 5. Department of Mathematics, College of General Education, Nagoya Univ. 1983.Google Scholar
10. Reimann, H.M. and Rychener, T., Funktionen beschrànkter minierer Oszillation, Lecture Notes in Math., Vol. 487, Springer-Verlag, Berlin Heidelberg, 1975.Google Scholar