In this paper, we establish a weighted version of the well-known Fréchet–Kolmogorov theorem, which holds for weights beyond \(A_\infty \). This weighted theory extends the previous known unsatisfactory results in the terms of relaxing the index to the natural range. As applications, we obtain the weighted compactness theory for the commutators of multilinear vector-valued Calderón–Zygmund type operators, including the commutators of multilinear Littlewood–Paley type operators. It is worthy to pointing out that the commutators we considered contain almost all the commutators formerly studied in this literature.

中文翻译：

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向量值多线性算子的加权Fréchet–Kolmogorov定理和紧性

在本文中，我们建立了著名的Fréchet–Kolmogorov定理的加权版本，该定理的权重超过\（A_ \ infty \）。该加权理论在将指标放宽到自然范围方面扩展了先前已知的不令人满意的结果。作为应用，我们获得了多线性矢量值Calderón–Zygmund型算子的换向器，包括多线性Littlewood–Paley型算子的换向器的加权紧致度理论。值得指出的是，我们考虑的换向器几乎包含了该文献中以前研究过的所有换向器。