ABSTRACT

In Koo and Wang (Joint Carleson measure and the difference of composition operators on $A_{\alpha }^p\left({\mathbf{B}}_n\right)$. J Math Anal Appl. 2014;419:1119–1142), the authors introduced a concept of joint Carleson measure and used it to characterize when the difference of two composition operators on weighted Bergman space over the unit ball is bounded or compact. In this paper, we extend the concept of joint Carleson measure to the polydisk setting and obtain analogue characterizations of the boundedness (compactness, resp.) of the difference of two composition operators on the weighted Bergman spaces over the unit polydisk, which may provide a unified approach for various ad hoc studies on the boundedness or the compactness of the difference of composition operators on polydisk. Moreover, we construct a concrete example to show that both the boundedness and the compactness depend on the index p when the dimension $n\ge 2$, which is in sharp contrast with the one-variable case where the boundedness and the compactness of the difference of two composition operators are independent of p>0. Due to the complexity of the Carleson measure on the unit polydisk, some new techniques are required in the polydisk setting.

摘要

在Koo和Wang（Joint Carleson测度以及组合算子在 ${\mathrm{一种}}_{\alpha }^p（{\mathbf{乙}}_{ñ}）$。J数学肛门应用。2014; 419：1119–1142），作者介绍了联合卡尔森测度的概念，并用它来刻画单位球上加权Bergman空间上的两个合成算子的差是有界的还是紧致的。在本文中，我们将联合Carleson测度的概念扩展到多盘设置，并获得了单位多盘加权Bergman空间上两个合成算子之差的有界性（紧致性，分别）的模拟特征，这可以提供统一方法，用于对多磁盘上成分算子的差异的有界性或紧密性进行各种专门研究。此外，我们构造了一个具体的例子来表明有界性和紧密性都取决于指标p当尺寸$ñ\ge 2$，这与两个变量算子的差的有界性和紧致性独立于p > 0的单变量情况形成鲜明对比。由于单元多磁盘上Carleson度量的复杂性，多磁盘设置中需要一些新技术。