Equivalent norms on Fock spaces with some application to extended Cesaro operators
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Abstract:
Let $F_\gamma ^p$ be the Fock space of all holomorphic functions $f$ in ${\mathbf C}^n$ with the Fock norm \[ \|f\|_{p, \gamma }=\left \{\int _{\mathbf {C}^n}\left |f(z)e^ {-\frac {\gamma |z|^2}{2}}\right |^{p}dA(z)\right \}^{\frac {1}{p}} <\infty , \] where $p, \gamma$ are positive numbers. We prove that, given any positive integer $m$, the Fock norm $\|f\|_{p, \gamma }$ is equivalent to \[ \sum _{|\alpha |\le m-1} |\partial ^\alpha f(0)|+ \left \{\sum _{|\alpha |= m} \int _{{\mathbf C}^n } \left | \partial ^\alpha f(z) (1+|z|)^{-m} e^ {-\frac {\gamma |z|^2}{2}} \right |^p dA(z)\right \}^{\frac 1p}.\] As some application we characterize these holomorphic functions $g$ in ${\mathbf C}^n$ for which the induced extended Cesaro operator $T_g$ is bounded (or compact) from one Fock space $F_\gamma ^p$ to another $F_\gamma ^q$.References
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Additional Information
- Zhangjian Hu
- Affiliation: Department of Mathematics, Huzhou Teachers College, Huzhou, Zhejiang, 313000, People’s Republic of China
- Email: huzj@hutc.zj.cn
- Received by editor(s): April 5, 2011
- Received by editor(s) in revised form: November 9, 2011
- Published electronically: April 24, 2013
- Additional Notes: This research was partially supported by the National Natural Science Foundation of China (11271124, 11101139), the Natural Science Foundation of Zhejiang Province (Y6090036, Y6100219) and the Foundation of Creative Group in Universities of Zhejiang Province (T200924).
- Communicated by: Richard Rochberg
- © Copyright 2013 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 141 (2013), 2829-2840
- MSC (2010): Primary 32A37; Secondary 47B38
- DOI: https://doi.org/10.1090/S0002-9939-2013-11550-9
- MathSciNet review: 3056573