| 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. |
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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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