Characterizations of Bergman spaces and Bloch space in the unit ball of $\textbf {C}^ n$
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- by Cai Heng Ouyang, Wei Sheng Yang and Ru Han Zhao PDF
- Trans. Amer. Math. Soc. 347 (1995), 4301-4313 Request permission
Abstract:
In this paper we prove that, in the unit ball $B$ of ${{\mathbf {C}}^n}$, a holomorphic function $f$ is in the Bergman space $L_a^p(B),\;0 < p < \infty$, if and only if \[ \int _B {|\tilde \nabla } f(z){|^2}|f(z){|^{p - 2}}{(1 - |z{|^2})^{n + 1}}d\lambda (z) < \infty ,\] where $\tilde \nabla$ and $\lambda$ denote the invariant gradient and invariant measure on $B$, respectively. Further, we give some characterizations of Bloch functions in the unit ball $B$, including an exponential decay characterization of Bloch functions. We also give the analogous results for $\operatorname {BMOA} (\partial B)$ functions in the unit ball.References
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Additional Information
- © Copyright 1995 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 347 (1995), 4301-4313
- MSC: Primary 32A37; Secondary 46E15
- DOI: https://doi.org/10.1090/S0002-9947-1995-1311908-X
- MathSciNet review: 1311908