Weighted estimates for the iterated commutators of multilinear maximal and fractional type operators

Volume 217 / 2013

Qingying Xue Studia Mathematica 217 (2013), 97-122 MSC: Primary 42B20; Secondary 42B25. DOI: 10.4064/sm217-2-1


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


  • Qingying XueSchool of Mathematical Sciences
    Beijing Normal University
    Laboratory of Mathematics and Complex Systems
    Ministry of Education
    Beijing, 100875, P.R. China

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