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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Hankel operators in the Bergman space and Schatten $p$-classes: The case $1<p<2$


Author: Jingbo Xia
Journal: Proc. Amer. Math. Soc. 129 (2001), 3559-3567
MSC (2000): Primary 47B10, 47B32, 47B35
Published electronically: May 21, 2001
MathSciNet review: 1860488
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Abstract:

K. Zhu proved in Amer. J. Math. 113 (1991), 147-167, that, for $2 \leq p < \infty $, the Hankel operators $H_{f}$ and $H_{\bar f}$ on the Bergman space belong to the Schatten class ${\mathcal{C}}_{p}$ if and only if the mean oscillation MO $(f)(z)= \{\widetilde {\vert f\vert^{2}}(z) - \vert\tilde f(z)\vert^{2}\}^{1/2}$ belongs to $L^{p}(D,(1-\vert z\vert^{2})^{-2}dA(z))$. In this paper we prove that the same result also holds when $1 < p < 2$.


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Additional Information

Jingbo Xia
Affiliation: Department of Mathematics, State University of New York at Buffalo, Buffalo, New York 14260
Email: jxia@acsu.buffalo.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-01-06217-7
PII: S 0002-9939(01)06217-7
Received by editor(s): April 11, 2000
Published electronically: May 21, 2001
Additional Notes: This work was supported in part by NSF grant DMS-9703515.
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2001 American Mathematical Society