Weighted norm inequalities for the commutators of multilinear singular integral operators

In this paper, the authors consider the weighted estimates for the commutators of multilinear Calderón-Zygmund operators. By introducing an operator which shifts the commutation, and establishing the weighted estimates for this new operator, the authors prove that, if p1∈(1,∞),p2,⋯,pm∈(1,∞],p∈(0,∞) with , then for any weight w, the commutators of m-linear Calderón-Zygmund operator are bounded from Lp1(ℝn,MLσ(log L)w)×Lp2(Rn,Mw)×⋯×Lpm(ℝn,Mw) to Lp(ℝn,w) with σ to be a constant depending only on p1 and the order of commutator.