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A ``deformation estimate" for the Toeplitz operators on harmonic Bergman spaces


Author: Congwen Liu
Journal: Proc. Amer. Math. Soc. 135 (2007), 2867-2876
MSC (2000): Primary 47B35, 47B38; Secondary 53D55
DOI: https://doi.org/10.1090/S0002-9939-07-08800-4
Published electronically: May 8, 2007
MathSciNet review: 2317963
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Abstract: Let $ B$ denote the open unit ball in $ \mathbb{R}^n$ for $ n\geq 2$ and $ dx$ the Lebesgue volume measure on $ \mathbb{R}^n$. For $ \alpha>-1$, the (weighted) harmonic Bergman space $ b^{2,\alpha}(B)$ is the space of all harmonic functions $ u$ which are in $ L^2(B,(1-\vert x\vert^2)^{\alpha}dx)$. For $ f\in L^{\infty}(B)$, the Toeplitz operator $ T_f^{(\alpha)}$ is defined on $ b^{2,\alpha}(B)$ by $ T_f^{(\alpha)}u = Q_{\alpha}[fu]$, where $ Q_{\alpha}$ is the orthogonal projection of $ L^2(B,(1-\vert x\vert^2)^{\alpha}dx)$ onto $ b^{2,\alpha}(B)$. In this note, we prove that for $ f\in C(B)\cap L^{\infty}(B)$ radial, $ \lim_{\alpha\to\infty} \Vert T_f^{(\alpha)}\Vert=\Vert f\Vert _{\infty}$.


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Additional Information

Congwen Liu
Affiliation: Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026 People’s Republic of China
Email: cwliu@nankai.edu.cn

DOI: https://doi.org/10.1090/S0002-9939-07-08800-4
Keywords: Toeplitz operators, harmonic Bergman spaces, Deformation Estimate, Berezin type transforms
Received by editor(s): November 30, 2005
Received by editor(s) in revised form: May 26, 2006
Published electronically: May 8, 2007
Additional Notes: This work was supported in part by the National Natural Science Foundation of China grant 10601025.
Dedicated: Dedicated to Professor Jihuai Shi on the occasion of his seventieth birthday.
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2007 American Mathematical Society

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