Geodesic
The Norm and Regularized Trace of the Cauchy Transform
Matematičeskie zametki, Tome 77 (2005) no. 6, pp. 844-853.

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In this paper, the norm of the Cauchy transform $C$ is obtained on the space $L^2(D,d\mu)$, where $d\mu=\omega(|z|)dA(z)$. Also, (for the case $\omega\equiv1$), the first regularized trace of the operator $C^*C$ on $L^2(\Omega)$ is obtained. The results are illustrated by examples, with different specific choices of the function $\omega$ and the domain $\Omega$. 
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M. R. Dostanic. The Norm and Regularized Trace of the Cauchy Transform. Matematičeskie zametki, Tome 77 (2005) no. 6, pp. 844-853. http://geodesic.mathdoc.fr/item/MZM_2005_77_6_a3/

[1] Anderson J. M., Hinkkanen A., “The Cauchy transform on bounded domain”, Proc. Amer. Math. Soc., 107 (1989), 179–185 | DOI | Zbl

[2] Gokhberg I. Ts., Krein M. G., Vvedenie v teoriyu lineinykh nesamosopryazhennykh operatorov, Nauka, M., 1965

[3] Watson G. N., A Treatise of the Theory of Bessel Functions, 2nd edition, Cambridge Univ. Press, Cambridge, 1962

[4] Beitman G., Erdeii A., Vysshie transtsendentnye funktsii, T. 2, Nauka, M., 1974

[5] Dostanić M. R., “Norm estimate of the Cauchy transform on $L^p(\Omega)$”, Integral Equations Operator Theory (to appear)

[6] Boyd D. W., “Best constants in a class of integral inequalities”, Pacific J. Math., 30:2 (1969), 367–383 | Zbl

[7] Dostanić M. R., “The properties of the Cauchy transform on a bounded domain”, J. Operator Theory, 36 (1996), 233–247 | Zbl

[8] Vekua I. N., Obobschennye analiticheskie funktsii, Nauka, M., 1988 | Zbl

[9] Dostanić M. R., “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 | DOI | Zbl

[10] Dostanić M. R., “Spectral properties of the operator or Riesz potential type”, Proc. Amer. Math. Soc., 126:8 (1998), 2291–2297 | DOI | Zbl

[11] Gorbachuk V. I., Gorbachuk M. L., Granichnye zadachi dlya differentsialnykh uravnenii s operatornymi koeffitsientami, Naukova dumka, Kiev, 1984 | Zbl

[12] Anderson J. M., Khavinson D., Lomonosov V., “Spectral properties of some integral operators arising in potential theory”, Quart. J. Math. (Oxford). Ser. 2, 1992, 387–407 | Zbl