$\mathbb{C}^{n}$中$\mu$-Bergman空间的刻画和微分复合算子

设$p>0$, $\mu$和$\mu_{1}$是$0,1)$上的正规函数. 本文首先给出了$\mathbb{C}^{n}$中单位球上$\mu$-Bergman空间$A^{p}(\mu)$的几种等价刻画; 然后 分别刻画了$A^{p}(\mu)$到$A^{p}(\mu_{1})$的 微分复合算子$D_{\varphi}$为有界算子以及紧算子的充要条件, 同时给出了当$p>1$时$D_{\varphi}$为 $A^{p}(\mu)$到$A^{p}(\mu_{1})$上紧算子的一种简捷充分条件和必要条件.

Characterizations and Differentiation Composition Operators of $\mu$-Bergman Space in $\mathbb{C}^n$

Let $p>0$, $\mu$ and $\mu_{1}$ be two normal functions on $0,1)$. In this paper, a kind of equivalent characterizations of the $\mu$-Bergman space on the unit ball in $\mathbb{C}^{n}$ are given first. Furthermore, the necessary and sufficient conditions that the differentiation composition operator $D_{\varphi}$ is a bounded operator or a compact operator from $A^{p}(\mu)$ to $A^{p}(\mu_{1})$ are given, respectively. At the same time, a simple sufficient condition and the necessary condition that $D_{\varphi}$ is a compact operator from $A^{p}(\mu)$ to $A^{p}(\mu_{1})$ are given.