Weighted norm inequalities for the commutators of multilinear singular integral operators

Abstract

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 p 1 ∈ ( 1 , ∞ ) , p 2 , ⋯ , p m ∈ ( 1 , ∞ ] , p ∈ ( 0 , ∞ ) with 1 / p = ∞ ∑ 1 ≤ k ≤ m 1 / p k , then for any weight w, the commutators of m-linear Calderón-Zygmund operator are bounded from L p 1 ( ℝ n , M L ( log L ) σ w ) × L p 2 ( R n , M w ) × ⋯ × L p m ( ℝ n , M w ) to L p ( ℝ n , w ) with σ to be a constant depending only on p 1 and the order of commutator.