加权Dirichlet空间上加权复合算子的有界性与紧性

Boundedness and Compactness of Weighted Composition Operators on Weighted Dirichlet Spaces

摘要: 本文研究了由双倍权(doubling weight)诱导的加权Dirichlet空间$D_{\omega }^2$ 上加权复合算子$W_{\phi ,\psi }$ 的有界性与紧性。我们首先回顾Liu和Lou (2013)在幂权Dirichlet空间$D_{\alpha }^2$ 上的经典结果，其利用Nevanlinna计数函数与Carleson测度给出了算子有界与紧的完整刻画。在此基础上，借助Chen (2023)关于双倍权Bergman空间上加权复合算子差分的Carleson测度理论，我们将Liu和Lou的结果推广到更一般的双倍权Dirichlet空间$D_{\omega }^2$ 中。我们建立了$W_{\phi ,\psi }$ 在$D_{\omega }^2$ 上有界(紧)的充分必要条件，这些条件分别用相关的拉回测度是(消失的) s-Carleson测度，以及Nevanlinna计数函数的加权积分增长行为来表述。本文的结果不仅扩展了经典幂权情形的理论，而且为研究双倍权Dirichlet空间上算子性质提供了新的工具。

Abstract: This paper investigates the boundedness and compactness of weighted composition operators $W_{\phi ,\psi }$ on weighted Dirichlet spaces $D_{\omega }^2$ induced by doubling weights. We first review the classical results of Liu and Lou (2013) on power-weighted Dirichlet spaces $D_{\alpha }^2$ , where they provided a complete characterization of operator boundedness and compactness using Nevanlinna counting functions and Carleson measures. Building on this foundation, we extend Liu and Lou’s results to the more general setting of doubling-weighted Dirichlet spaces $D_{\omega }^2$ by leveraging Chen’s (2023) theory of Carleson measures for differences of weighted composition operators on doubling-weighted Bergman spaces. We establish necessary and sufficient conditions for the boundedness (compactness) of $W_{\phi ,\psi }$ on $D_{\omega }^2$ , which are formulated in terms of the pullback measures being (vanishing) s-Carleson measures and the weighted integral growth behavior of Nevanlinna counting functions. The results of this paper not only extend the classical theory for power-weighted cases but also provide new tools for studying operator properties on doubling-weighted Dirichlet spaces.