Mathematical Notes

, Volume 77, Issue 5–6, pp 777–786 | Cite as

The Norm and Regularized Trace of the Cauchy Transform

  • M. R. Dostanic


In this paper, the norm of the Cauchy transform C is obtained on the space L2(D, dμ), where dμ = ω(|z|) dA(z). Also, (for the case ω ≡ 1), the first regularized trace of the operator C* C on L2(Ω) is obtained. The results are illustrated by examples, with different specific choices of the function ω and the domain Ω.

Key words

Cauchy transform regularized trace operator function space boundary-value problem Bessel function Parseval’s equality 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    J. M. Anderson and A. Hinkkanen, “The Cauchy transform on bounded domain,” Proc. Amer. Math. Soc., 107 (1989), 179–185.Google Scholar
  2. 2.
    I. Ts. Gokhberg and M. G. Krein, Introduction to the Theory of Linear Non-Self-Adjoint Operators [in Russian], Nauka, Moscow, 1965.Google Scholar
  3. 3.
    G. N. Watson, A Treatise of the Theory of Bessel Functions, 2nd edition, Cambridge Univ. Press, Cambridge, 1962.Google Scholar
  4. 4.
    H. Bateman and A. Erdelyi, Higher Transcendental Functions, vol. 2, McGraw-Hill, New York-London, 1953; Russian translation: Nauka, Moscow, 1974.Google Scholar
  5. 5.
    M. R. Dostanic, “Norm estimate of the Cauchy transform on L p(Ω),” Integral Equations Operator Theory (to appear).Google Scholar
  6. 6.
    D. W. Boyd, “Best constants in a class of integral inequalities,” Pacific J. Math., 30 (1969), no. 2, 367–383.Google Scholar
  7. 7.
    M. R. Dostanic, “The properties of the Cauchy transform on a bounded domain,” J. Operator Theory, 36 (1996), 233–247.Google Scholar
  8. 8.
    I. N. Vekua, Generalized Analytic Functions [in Russian], Nauka, Moscow, 1988.Google Scholar
  9. 9.
    M. R. Dostanic, “Spectral properties of the Cauchy operator and its product with Bergman’s projection on a bounded domain,” Proc. London Math. Soc. (3), 76 (1998), 667–684.CrossRefGoogle Scholar
  10. 10.
    M. R. Dostanic, “Spectral properties of the operator or Riesz potential type,” Proc. Amer. Math. Soc., 126 (1998), no. 8, 2291–2297.CrossRefGoogle Scholar
  11. 11.
    V. I. Gorbachuk and M. L. Gorbachuk, Boundary-Value Problems for the Differential Equations with Operator Coefficients [in Russian], Naukova Dumka, Kiev, 1984.Google Scholar
  12. 12.
    J. M. Anderson, D. Khavinson, and V. Lomonosov, “Spectral properties of some integral operators arising in potential theory,” Quart. J. Math. (Oxford). Ser. 2 (1992), 387–407.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • M. R. Dostanic
    • 1
  1. 1.Mathematics Dept.Belgrade UniversitySerbia and MontenegroYugoslavia

Personalised recommendations