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

In this paper, the necessity of BMO for boundedness of commutators of multilinear singular integral operators on weighted Lebesgue spaces is investigated. The results relax the restriction on the weight class to general multiple weights, which can be regarded as an essential improvement of the result of Chaffee and Cruz-Uribe (2018) and Guo, Lian and Wu (2020). Our approach elaborates on a commonly used method of expanding the kernel locally in Fourier series, recovering many known results but yielding also numerous new ones. In particular, we answer the question about the necessity issue for iterated commutators of multilinear singular integral operators.