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 $Z^+\times \{1,\dots,m\}$ and denote $d\vec{y}=dy_1\cdots dy_m$. Define the commutator $T_{\vec{b},S}$ of $T$, and $T^*_{\vec{b},S}$ of $T^*$ by $T_{\vec{b},S}(\vec{f})(x)= \int_{\mathbb{R}^{nm}}\prod_{(i,j)\in S}(b_i(x)-b_i(y_j)) K(x,y_{1},\dots,y_{m}) \prod_{j=1}^mf_j(y_j)d\vec{y}$ and $T^*_{\vec{b},S}(\vec{f})(x)= \sup_{\delta>0}\big|\int_{{\sum_{j=1}^m|x-y_j|^2>\delta^2}} \prod_{(i,j)\in S}(b_i(x)-b_i(y_j))K(x,y_{1},\dots,y_{m}) \prod_{j=1}^mf_{j}(y_j)d\vec{y}\big|$. 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.