张学军,黎深莲.Cn中单位球上$mu$-Bloch函数的等价刻画[J].数学年刊A辑,2016,37(4):367~376
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:
中文关键词:  $\mu$-Bloch空间, 等价刻画, 单位球
英文关键词:$\mu$-Bloch space, Equivalent characterization, Unit ball
基金项目:本文受到国家自然科学基金 (No.11571104), 湖南省自然科学基金(No.2015JJ2095), 湖南省重点学科建设项目和湖南师范大学数学与计算机科学学院 高性能计算与随机信息处理省部共建教育部重点实验室的资助.
Author NameAffiliation
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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中文摘要:
      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.
英文摘要:
      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 $B$.
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