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

TL;DR

This paper develops a weighted compactness theory for vector-valued multilinear Calderón-Zygmund operators by establishing a generalized weighted Fréchet-Kolmogorov theorem, extending previous results to broader weights and operator classes.

Contribution

It introduces a weighted Fréchet-Kolmogorov theorem applicable to a wider class of weights and extends compactness results to various multilinear operators and their commutators.

Findings

Established a weighted Fréchet-Kolmogorov theorem for weights beyond $A_$.

Proved weighted compactness for commutators of multilinear Calderón-Zygmund operators.

Extended results to commutators of multilinear Littlewood-Paley operators.

Abstract

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 $A_\infty$. 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.