The necessity theory for commutators of multilinear singular integral operators: the weighted case

TL;DR

This paper develops a necessity theory for commutators of multilinear singular integral operators on weighted Lebesgue spaces, broadening the class of weights considered and providing new insights into their iterated commutators.

Contribution

It extends the necessity theory to general multiple weights and addresses the necessity of iterated commutators, improving upon previous restrictions and results.

Findings

Broadened weight class to general multiple weights

Established necessity conditions for iterated commutators

Provided new results on the structure of multilinear singular integrals

Abstract

In this paper, the necessity theory for commutators of multilinear singular integral operators on weighted Lebesgue spaces is investigated. The results relax the restriction of the weights class to the general multiple weights, which can be regarded as an essential improvement of \cite{ChafCruz2018,GLW2020}. Our approach elaborates on a commonly expanding the kernel locally by Fourier series, recovering many known results but yielding also numerous new ones. In particular, we answer the question about the necessity theory of the iterated commutators of the multilinear singular integral operators.