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数学进展 - 2016, Vol. 45(2): 263-270
研究论文
单位球上Bloch型空间到F(p,q,s)空间的Volterra复合算子
Volterra Composition Operators From Bloch-type Spaces to \mbox{\boldmath $F(p,q,s)$} Spaces on the Unit Ball

何忠华1,曹广福2,何莉2
HE Zhonghua1,CAO Guangfu2, HE Li2

1. 广东金融学院应用数学系, 广州, 广东, 510521;
2. 广州大学数学与信息科学学院, 广州, 广东, 510006
1. Department of Applied Mathematics, Guangdong University of Finance, Guangzhou,Guangdong, 510521, P. R. China;
2. School of Mathematics and Information Sciences, Guangzhou University, Guangzhou, Guangdong, 510006,P. R. China

收稿日期: 2014-08-04
出版日期: 2016-03-10
2016, Vol. 45(2): 263-270
DOI: 10.11845/sxjz.2014118b


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摘要 记 $H(\mathbb{B})$ 为 $\mathbb{C}^n$ 中单位球 $\mathbb{B 上的解析函数空间. 设 $\varphi$ 为 $\mathbb{B 到自身的解析映射, $g\in H(\mathbb{B})$, $\mu$ 为正规权, 定义 Volterra 复合算子为$$(V^g_{\varphi}f)(z)=\int^1_0f(\varphi(tz))\mathcal{R}g(tz)\frac{{\rm d}t}{t}.$$本文考虑 Volterra 复合算子 $V^g_{\varphi 从 $\mathcal{B}_{\mu 或 $\mathcal{B}_{\mu,0 空间到 $F(p,q,s)$ 或 $F_0(p,q,s)$ 空间上的有界性和紧性, 得出了算子 $V^g_{\varphi : $\mathcal{B}_{\mu}(\mathcal{B}_{\mu,0})\rightarrow F(p,q,s)$ 或 $\mathcal{B}_{\mu}(\mathcal{B}_{\mu,0})\rightarrow F_{\rm 0}(p,q,s)$ 的紧性与有界性等价. 同时,也给出了算子 $V^g_{\varphi 从 $\mathcal{B}^{\alpha 或 $\mathcal{B}^{\alpha}_0$ 空间到 $F(p,q,s)$ 或 $F_{0}(p,q,s)$ 空间上的紧性和有界性刻画.
Abstract:Let $H(\mathbb{B})$ denote the space of all holomorphic functions on the unit ball $\mathbb{B of $\mathbb{C}^n$. Let $\varphi$ be a holomorphic self-map of $\mathbb{B and $g\in H(\mathbb{B})$. In this paper, we investigate the boundedness and compactness of the Volterra composition operator $$(V^g_{\varphi}f)(z)=\int^1_0f(\varphi(tz))\mathcal{R}g(tz)\frac{\textrm{d}t}{t, $$which maps from Bloch-type spaces $\mathcal{B}_{\mu or $\mathcal{B}_{\mu,0, where $\mu$ is a normal weight, to $F(p,q,s)$ or $F_0(p,q,s)$ spaces on $\mathbb{B. We obtain that the compactness and boundedness of the operator $V^g_{\varphi}: \mathcal{B}_{\mu}( \mathcal{B}_{\mu,0} )\rightarrow F(p,q,s)$ or $\mathcal{B}_{\mu}(\mathcal{B}_{\mu,0})\rightarrow F_0(p,q,s)$ are equivalent. Also, we characterize the boundedness and compactness of the Volterra operator $V^g_{\varphi which maps from $\mathcal{B}^{\alpha or $\mathcal{B}^{\alpha}_0$ to $F(p,q,s)$ or $F_0(p,q,s)$.
PACS:  O177.2  
基金资助:基金项目:国家自然科学基金 (No. 11271092), 广东省重点学科建设资助和广东金融学院重点学科建设项目资助.
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