$mathbb{C}^{n}$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 $[0,1)$. Inthis 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.