Weighted Estimates for the iterated Commutators of Multilinear Maximal and Fractional Type Operators

In this paper, the following iterated commutators $T_{*,\Pi b}$ of maximal operator for multilinear singular integral operators and $I_{\alpha, \Pi b}$ of multilinear fractional integral operator are introduced and studied \aligned T_{*,\Pi b}(\vec{f})(x)&=\sup_{\delta>0}\bigg|[b_1,[b_2,...[b_{m-1},[b_m,T_\delta]_m]_{m-1}...]_2]_1 (\vec{f})(x)\bigg|, \aligned I_{\alpha, \Pi b}(\vec{f})(x)&=[b_1,[b_2,...[b_{m-1},[b_m,I_\alpha]_m]_{m-1}...]_2]_1 (\vec{f})(x), where $T_\delta$ are the smooth truncations of the multilinear singular integral operators and $I_{\alpha}$ is the multilinear fractional integral operator, $b_i\in BMO$ for $i=1,...,m$ and $\vec {f}=(f_1,...,f_m)$. Weighted strong and $L(\log L)$ type end-point estimates for the above iterated commutators associated with two class of multiple weights $A_{\vec{p}}$ and $A_{(\vec{p}, q)}$ are obtained, respectively.