This paper is concerned with the pointwise estimates for the sharp function of two kinds of maximal commutators of multilinear singular integral operators T∑b^＊ and TПb^＊ which are generalized by a weighted BMO function b and a multilinear singular integral operator T, respectively. As applications, some commutator theorems are established.
In this note, the author prove that maximal Bocher-Riesz commutator Bδ,＊^b generated by operator Bδ,＊ and function b∈ BMO（ω） is a bounded operator from L^p（μ） into L^p（ν）, where w∈ （μν^- 1）^1/p,μ,v ∈ Ap for 1 〈 p 〈 ∞. The proof relies heavily on the pointwise estimates for the sharp maximal function of the commutator Bδ,＊^b.
