Abstract
We prove two density theorems for quadrature domains in \(\mathbb {C}^n\), \(n \ge 2\). It is shown that quadrature domains are dense in the class of all product domains of the form \(D \times \Omega \), where \(D \subset \mathbb {C}^{n-1}\) is a smoothly bounded domain satisfying Bell’s Condition R and \(\Omega \subset \mathbb {C}\) is a smoothly bounded domain and also in the class of all smoothly bounded complete Hartogs domains in \(\mathbb {C}^2\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Recall that \(K_G(z, w)\) is conjugate holomorphic in \(w\) and hence this should not cause any confusion with the notation for mixed partial derivatives introduced just after (1.1). In any case, what is meant will be clear from the functions that are involved.
References
Aharonov, D., Shapiro, H.S.: Domains on which analytic functions satisfy quadrature identities. J. D’Anal. Math. 30, 39–73 (1976)
Bell, S.: Non-vanishing of the Bergman kernel function at boundary points of certain domains in \(\mathbb{C}^n\). Math. Ann. 244, 69–74 (1979)
Bell, S.: Proper holomorphic mappings between circular domains. Comm. Math. Helv. 57, 532–538 (1982)
Bell, S.: The Cauchy Transform, Potential Theory and Conformal Mapping. CRC Press, Boca Raton (1992)
Bell, S.: Algebraic Mappings of Circular Domains in \(\mathbb{C}^n\), Several Complex Variables (Stockholm, 1987–1988). Mathematical Notes, vol. 38, pp. 126–135. Princeton University Press, Princeton (1993)
Bell, S.: Density of quadrature domains in one and several complex variables. Complex Var. Elliptic Equ. 54(3–4), 165–171 (2009)
Bell, S., Boas, H.: Regularity of the Bergman projection in weakly pseudoconvex domains. Math. Ann. 257, 23–30 (1981)
Boas, H., Straube, E.: Complete Hartogs domains in \(\mathbb{C}^2\) have regular Bergman and Szegö projections. Math. Z. 201, 441–454 (1989)
Catlin, D.: Subelliptic estimates for the \({\overline{\partial }}\)-Neumann problem on pseudoconvex domains. Ann. Math. 126, 131–191 (1987)
Chakrabarti, D., Shaw, M.-C.: The Cauchy–Riemann equations on product domains. Math. Ann. 977–998 (2011)
Ebenfelt, P., Gustafsson, B., Khavinson, D., Putinar, M.: Quadrature Domains and Their Applications. Operator Theory: Advances and Applications, vol. 156. Birkhäuser, Basel (2005)
Gustafsson, B.: Quadrature domains and the Schottky double. Acta Appl. Math. 1, 209–240 (1983)
Horvàth, J.: Topological Vector Spaces and Distributions, vol. I. Addison-Wesley, Reading (1966)
Kohn, J.J.: Harmonic integrals on strongly pseudoconvex manifolds I. Ann. Math. 78, 112–148 (1963)
Kohn, J.J.: Harmonic integrals on strongly pseudoconvex manifolds II. Ann. Math. 79, 450–472 (1964)
Sakai, M.: Regularity of a boundary having a Schwarz function. Acta Math. 166, 263–297 (1991)
Shapiro, H.S.: The Schwarz Function and Its Generalization to Higher Dimensions. University of Arkansas Lecture Notes in the Mathematical Sciences, vol. 9. Wiley, New York (1992)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Dmitry Khavinson.
P. Haridas was supported by a UGC–CSIR Grant. K. Verma was supported by the DST SwarnaJayanti Fellowship 2009–2010 and a UGC–CAS Grant.
Rights and permissions
About this article
Cite this article
Haridas, P., Verma, K. Quadrature Domains in \({{\mathbb C}}^n\) . Comput. Methods Funct. Theory 15, 125–141 (2015). https://doi.org/10.1007/s40315-014-0090-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40315-014-0090-y