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

Abstract

In this paper, the following iterated commutators $T_{\ast ,\Pi b}$ of maximal operator for multilinear singular integral operators and $I_{\alpha ,\Pi b}$ of multilinear fractional integral operator are introduced and studied $$\aligned T_{*,Πb}(\vec{f})(x)&=\sup_{δ>0}\bigg|[b_1,[b_2,...[b_{m-1},[b_m,T_δ]_m]_{m-1}...]_2]_1 (\vec{f})(x)\bigg|,$$ $$\aligned I_{α, Πb}(\vec{f})(x)&=[b_1,[b_2,...[b_{m-1},[b_m,I_α]_m]_{m-1}...]_2]_1 (\vec{f})(x),$$ 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.

23 pages, Corrected typos

References in corpus (2)

• The multilinear strong maximal function
• Weighted inequalities for multilinear potential operators and its commutators