A “deformation estimate" for the Toeplitz operators on harmonic Bergman spaces
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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 $L^2(B,(1-|x|^2)^{\alpha }dx)$. 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 $L^2(B,(1-|x|^2)^{\alpha }dx)$ onto $b^{2,\alpha }(B)$. In this note, we prove that for $f\in C(B)\cap L^{\infty }(B)$ radial, $\lim _{\alpha \to \infty } \|T_f^{(\alpha )}\|=\|f\|_{\infty }$.References
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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
- 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.
- Communicated by: Joseph A. Ball
- © Copyright 2007 American Mathematical Society
- 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
- MathSciNet review: 2317963
Dedicated: Dedicated to Professor Jihuai Shi on the occasion of his seventieth birthday.