Joint Carleson measure for the difference of composition operators on the polydisks

Abstract

In Koo and Wang (Joint Carleson measure and the difference of composition operators on A(alpha)(p)(B-n). 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 = 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.