The purpose of this paper is to study sparse domination estimates of composition operators in the setting of complex function theory. The method originates from proofs of the $A_2$ theorem for Calder\'on-Zygmund operators in harmonic analysis. Using this tool from harmonic analysis, some new characterizations are given for the boundedness and compactness of weighted composition operators acting between weighted Bergman spaces in the upper half plane. Moreover, we establish a new weighted type estimate for the holomorphic Bergman-class functions, for a new class of weights, which is adapted to Sawyer--testing conditions. We also extend our results to the unit ball $\mathbb B$ in $\mathbb C^n$.

中文翻译：

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加权伯格曼空间上加权复合算子的稀疏支配

本文的目的是研究复函数理论背景下复合算子的稀疏支配估计。该方法源于调和分析中 Calder\'on-Zygmund 算子的 $A_2$ 定理的证明。使用来自调和分析的这个工具，对作用在上半平面加权 Bergman 空间之间的加权合成算子的有界性和紧致性给出了一些新的表征。此外，我们为全纯伯格曼类函数建立了一个新的加权类型估计，用于新的权重类，它适用于索耶测试条件。我们还将我们的结果扩展到 $\mathbb C^n$ 中的单位球 $\mathbb B$。