Hostname: page-component-8448b6f56d-t5pn6 Total loading time: 0 Render date: 2024-04-18T08:23:17.904Z Has data issue: false hasContentIssue false
  • English
  • Français

L2-boundedness of the cauchy transform on smooth non-Lipschitz curves

Published online by Cambridge University Press:  22 January 2016

Hyeonbae Kang  and
Jin Keun Seo
Hyeonbae Kang
Affiliation:
Department of Mathematics, Soong Sil University, Sangdo-Dong, Dongjak-Gu, Seoul, 156-743, Korea
Jin Keun Seo
Affiliation:
GARC, Seoul National University, Shinlim-Dong, Kwanak-Gu, Seoul, 135-110, Korea, Department of Mathematics, POSTECH, Pohang, Korea
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let Γ be a curve defined by y = A(x) in R2. The Cauchy transform on the curve Γ is a singular integral operator defined by the singular integral kernel

(1.1)

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 130 , June 1993 , pp. 123 - 147
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1993

References

[Cal1] Calderón, A. P., Cauchy integrals on Lipschitz curves and related operators, Proc. Nat. Acad. Sci. USA, 74 (1977), 1324–1327.CrossRefGoogle ScholarPubMed
[Cal2] Calderón, A. P., Commutators, Singular integrals on Lipschitz curves and applications, Proceedings of ICM (Helsinki, 1978), Acad. Sci. Fennica, Helsinki (1980), 85–96.Google Scholar
[Chr] Chris, M., Lectures on Singular Integral Operators, CBMS 77, Amer. Math. Soc, 1990.Google Scholar
[C.D.M] Coifman, R. R., David, G., Meyer, Y., La solution des conjectures de Calderón, Advances in Math., 48 (1983), 144–148.CrossRefGoogle Scholar
[C.J.S] Coifman, R. R., Jones, P., and Semmes, S., Two elementary proofs of the L-boundedness of Cauchy integrals on Lipschiz curves, J. Amer. Math. Soc, 2 (1989), 553–564.Google Scholar
[C.M.M] Coifman, R. R., McIntosh, A., Meyer, Y., L’intégrale de Cauchy definit un opérateur bornée sur L pour courbes lipschitziennes, Ann. of Math., 116 (1982), 361–387.CrossRefGoogle Scholar
[CM] Coifman, R. R. and Meyer, Y., Au dela des opérateurs pseudo-differentielss, Asterique 57, Societe Mathematique de France, 1978.Google Scholar
[DJ] David, G. and Journé, J.-L., A boundedness criterion for generalized Calderón-Zygmund operators, Ann. of Math., 120 (1984), 371–397.CrossRefGoogle Scholar
[Jou] Journé, J.-L., Calderón-Zygmund operators, pseudo-differential operators, and the Cauchy integral of Calderón, Lecture Notes in Math., 994, Springer Verlag, New York, 1983.Google Scholar
[Mur] Murai, T., A real variable method for the Cauchy transform, and analytic capacity, Lecture Note in Math., 1307, Springer-Verlag, New York, 1988.Google Scholar
[Stl] Stein, E., Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, 1970.Google Scholar
[St2] Stein, E., Beijing Lectures in harmonic analysis, Princeton Univ. Press Princeton, 1986.Google Scholar
[Tor] Torchinsky, A., Real-variable methods in Harmonic Analysis, Academic Press, 1986.Google Scholar