Weighted Fréchet–Kolmogorov Theorem and Compactness of Vector-Valued Multilinear Operators

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