Generalized commutators of multilinear Calderón–Zygmund type operators

Abstract

Let T be an m-linear Calderón–Zygmund operator with kernel K and T^* be the maximal operator of T. Let S be a finite subset of Z⁺ × {1,…,m} and denote d$\vec{y}$ = dy₁ … dy_m. Define the commutator T_$\vec{b}$,S of T, and T^*_$\vec{b}$,S of T^* by T_$\vec{b}$,S($\vec{f}$)(x) = ∫_ℝ^nm ∏_(i,j)∈S(b_i(x) − b_i(y_j)) K(x,y₁,…,y_m) ∏_j=1^m f_j(y_j)d$\vec{y}$ and T^*_$\vec{b}$,S($\vec{f}$)(x) = sup_δ>0 | ∫_{∑j=1^m |x−yj|² > δ²} ∏_(i,j)∈S(b_i(x) − b_i(y_j))K(x,y₁,…,y_m) ∏_j=1^m f_j(y_j)d$\vec{y}$ |. These commutators are reflexible enough to generalize several kinds of commutators which already existed. We obtain the weighted strong and endpoint estimates for T_$\vec{b}$,S and T^*_$\vec{b}$,S with multiple weights. These results are based on an estimate of the Fefferman–Stein sharp maximal function of the commutators, which is proved in a pretty much more organized way than some known proofs. Similar results for the commutators of vector-valued multilinear Calderón–Zygmund operators are also given.