In this work, we provide a complete characterization of the boundedness of two classes of multiparameter Forelli–Rudin type operators from one mixed-norm Lebesgue space \(L^{{\vec {p}}}\) to another space \(L^{{\vec {q}}}\), when \(1\le \vec {p}\le {\vec {q}}<\infty \), equipped with possibly different weights. Using these characterizations, we establish the necessary and sufficient conditions for both \(L^{{\vec {p}}}-L^{{\vec {q}}}\) boundedness of the weighted multiparameter Berezin transform and \(L^{{\vec {p}}}-A^{{\vec {q}}}\) boundedness of the weighted multiparameter Bergman projection, where \(A^{{\vec {q}}}\) denotes the mixed-norm Bergman space. Our approach presents several novelties. Firstly, we conduct refined integral estimates of holomorphic functions on the unit ball in \({\mathbb {C}}^n\). Secondly, we adapt the classical Schur’s test to different weighted mixed-norm Lebesgue spaces. These improvements are crucial in our proofs and allow us to establish the desired characterization and sharp conditions.

在这项工作中，我们提供了两类多参数 Forelli–Rudin 类型算子从一个混合范数勒贝格空间\(L^{{\vec {p}}}\)到另一个空间\(L ^{{\vec {q}}}\)，当\(1\le \vec {p}\le {\vec {q}}<\infty \)时，配备可能不同的权重。利用这些特征，我们为加权多参数 Berezin 变换的L^{{\vec {p}}}-L^{{\vec {q}}}\) 有界性和\( L^{{\vec {p}}}-A^{{\vec {q}}}\) 加权多参数伯格曼投影的有界性，其中\(A^{{\vec {q}}}\)表示混合范数伯格曼空间。我们的方法提出了一些新颖之处。首先，我们对\({\mathbb {C}}^n\)中的单位球上的全纯函数进行精细积分估计。其次，我们将经典的 Schur 检验应用于不同的加权混合范数勒贝格空间。这些改进对于我们的证明至关重要，使我们能够建立所需的表征和锐利条件。