Abstract
We found the exact asymptotics of the singular numbers for the Cauchy transform and its product with Bergman’s projection over the space \(L^{2}(\Omega ),\) where \(\Omega \) is a doubly-connected domain in the complex plane.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Data Availability
The data that support the findings of this study are available from the corresponding author on request.
References
Anderson, J.M., Hinkkanen, A.: The Cauchy transform on bounded domain. Proc. Amer. Math. Soc. 107, 179–185 (1989)
Anderson, J.M., Khavinson, D., Lomonosov, V.: Spectral properties of some integral operators arising in potential theory. Quart. J. Math. Oxford 43(2), 387–407 (1992)
Arazy, J., Khavinson, D.: Spectral estimates of Cauchy’s transform in \(L^{2}(\Omega )\). Integral Equ. Oper. Theory. 15, 901–919 (1992)
Bell, S.R.: The Cauchy Transform, Potential Theory and Conformal Mapping, 2nd Edition, Purdue University West Lafayette, Indiana. Taylor and Francis Group, USA (2015)
Bergman, S.: The kernel function and conformal mapping, Mathematical Surveys 5. Amer. Math. Soc., New York (1950). Reprinted 1970. MR 12, 402a Zbl 0040.19001
Birman, M.S., Solomjak, M.Z.: Estimates of singular values of the integral operators. Uspekhi Mat. Nauk T32(193), 17–84 (1977)
Dostanić, M.: Estimate of the second term in the spectral asymptotic of Cauchy transform. J. Funct. Anal. 249, 55–74 (2007)
Dostanić, M.: The properties of the Cauchy transform on a bounded domain. J. Oper. Theory 36, 233–247 (1996)
Dostanić, M.: Spectral properties of the Cauchy operator and its product with Bergmans’s projection on a bounded domain. Proc. London Math. Soc. 76, 667–684 (1998)
Dostanić, M.: Cauchy operator on Bergman space of harmonic functions on unit disc. Matematicki Vesnik, 63-67 (2010)
Dostanić, M.: Norm Estimate of the Cauchy Transform on \(L^{p}(\Omega )\). Integral Equ. Oper. Theory 54, 465–475 (2005)
Gohberg, I.C., Krein, M.G.: Introduction to the Theory of Linear Nonselfadjoint Operators, Translations of Mathematical Monographs 18. American Mathematical Society, Providence, RI (1969)
Goluzin, G.M.: Geometric theory of functions of a complex variable. American Mathematical society, Vol. 26
Jeong, M., Mityushev, V.: The Bergman kernel for circular multiply connected domains. Pacific J. Math. 233(1) November (2007)
Kalaj, D., Melentijević, P., Zhu, J.-F.: Lp-theory for Cauchy-transform on the unit disk. J. Funct. Anal. 282(4), 109337 (2022)
Krantz, S.: A tale of three kernels, Complex Var. Elliptic Equ. 53(11) (2008)
Nehari, Z.: Conformal mapping, Dover Publications. INC, New York (2012)
O’Brien, D.M.: A Simple test for nuclearity of integral operators on \(L_{2}(\mathbb{R} ^{n})\). Austral Math. Soc. (Series A) 33, 193–196 (1982)
Range, R.M.: Holomorphic functions and integral representations in several complex variables. Springer, Berlin (1986)
Takač, P.: On \(l^{p}\) summability of the characteristic values of integral operators on \(L^{2}(\mathbb{R} ^{n}\). Integral Equ. Oper. Theory 10, 819–840 (1987)
Vujadinović, D.J.: Spectral estimates of Cauchy’s operator on Bergman space of harmonic functions. J. Math. Anal. Appl. 437(2), 902–911 (2016)
Vujadinović, D.J.: Spectral asymptotic of Cauchy’s operator on harmonic Bergman space for a simply connected domain. Complex Var. Elliptic Equ. 63(6), 770–782 (2018)
Author information
Authors and Affiliations
Contributions
There is only one author.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Vujadinović, D. Spectral Asymptotics of the Cauchy Operator and its Product with Bergman’s Projection on a Doubly Connected Domain. Potential Anal (2024). https://doi.org/10.1007/s11118-024-10139-3
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11118-024-10139-3