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The canonical solution operator to $\overline{\partial}$ restricted to Bergman spaces


Author: Friedrich Haslinger
Journal: Proc. Amer. Math. Soc. 129 (2001), 3321-3329
MSC (2000): Primary 32W05; Secondary 32A36
DOI: https://doi.org/10.1090/S0002-9939-01-05953-6
Published electronically: April 2, 2001
MathSciNet review: 1845009
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Abstract:

We first show that the canonical solution operator to $\overline{\partial}$restricted to $(0,1)$-forms with holomorphic coefficients can be expressed by an integral operator using the Bergman kernel. This result is used to prove that in the case of the unit disc in $\mathbb C $ the canonical solution operator to $\overline{\partial}$restricted to $(0,1)$-forms with holomorphic coefficients is a Hilbert-Schmidt operator. In the sequel we give a direct proof of the last statement using orthonormal bases and show that in the case of the polydisc and the unit ball in $\mathbb C^n, n>1,$ the corresponding operator fails to be a Hilbert-Schmidt operator. We also indicate a connection with the theory of Hankel operators.


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Additional Information

Friedrich Haslinger
Affiliation: Institut für Mathematik, Universität Wien, Strudlhofgasse 4, A-1090 Wien, Austria
Email: friedrich.haslinger@univie.ac.at

DOI: https://doi.org/10.1090/S0002-9939-01-05953-6
Keywords: $\overline\partial$-equation, Bergman kernel
Received by editor(s): March 20, 2000
Published electronically: April 2, 2001
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2001 American Mathematical Society

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