Cn中单位球上$mu$-Bloch函数的等价刻画
Equivalent Characterizations of $\mu$-Bloch Functions on the Unit Ball in ${\bf C}^{\it n}$
Received: May 26, 2015  Revised: December 14, 2015
DOI：

 Author Name Affiliation ZHANG Xuejun College of Mathematics and Computer Science, Hunan Normal University, Changsha 410006, China. LI Shenlian College of Mathematics and Computer Science, Hunan Normal University, Changsha 410006, China.
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设$\mu$是$[0,1)$上的正规函数, 给出了${\bf C}^{\it n}$中单位球$B$上$\mu$-Bloch空间$\beta_{\mu}$中函数的几种刻画. 证明了下列条件是等价的: (1) $f\in \beta_{\mu}$; \ (2) $f\in H(B)$且函数$\mu(|z|)(1-|z|^{2})^{\gamma-1}R^{\alpha,\gamma}f(z)$ 在$B$上有界; (3) $f\in H(B)$ 且函数${\mu(|z|)(1-|z|^{2})^{M_{1}-1}\frac{\partial^{M_{1}} f}{\partial z^{m}}(z)}$ 在$B$上有界, 其中$|m|=M_{1}$; (4) $f\in H(B)$ 且函数${\mu(|z|)(1-|z|^{2})^{M_{2}-1}R^{(M_{2})}f(z)}$ 在$B$上有界.
Let $\mu$ be a normal function on $[0,1)$. In this paper, the authors give some equivalent characterizations of $\mu$-Bloch functions on the unit ball in ${\bf C}^{\it n}$. They prove that the following conditions are equivalent: (1) \ $f\in \beta_{\mu}$; (2) \ $f\in H(B)$ and the function $\mu(|z|)(1-|z|^{2})^{\gamma-1}R^{\alpha,\gamma}f(z)$ is bounded in $B$; (3) \ $f\in H(B)$ and the function ${\mu(|z|)(1-|z|^{2})^{M_{1}-1}\frac{\partial^{M_{1}} f}{\partial z^{m}}(z)}$ is bounded in $B$, where $|m|=M_{1}$; (4) \ $f\in H(B)$ and the function ${\mu(|z|)(1-|z|^{2})^{M_{2}-1}R^{(M_{2})}f(z)}$ is bounded in $B$.