Weighted Fr\'{e}chet-Kolmogorov theorem and compactness of vector-valued multilinear operators

In this paper, we gave a weighted compactness theory for the generalized commutators of vecotor-valued multilinear Calder'{o}n-Zygmund operators. This was done by establishing a weighted Fr'{e}chet-Kolmogorov theorem, which holds for weights not merely in $A_{\mathrm{\infty }}$. This weighted theory also extends the previous known unsatisfactory results in the terms of relaxing the index to the natural range. As consequences, we not only obtained the weighted compactness theory for the commutators of multilinear Calder'{o}n-Zygmund operators, but also extended the same results to the commutators of multilinear Littlewood-Paley type operators. In addition, the generalized commutators contain almost all the commutators formerly considered in this literature.