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 ${\bf C}^{\it n}$

DOI：

 作者 单位 张学军 湖南师范大学数学与计算机科学学院, 长沙 410006. 黎深莲 湖南师范大学数学与计算机科学学院, 长沙 410006.

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 $B$.