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

Qingying Xue ¹

¹ School of Mathematical Sciences Beijing Normal University Laboratory of Mathematics and Complex Systems Ministry of Education Beijing, 100875, P.R. China

Studia Mathematica, Tome 217 (2013) no. 2, pp. 97-122

Résumé

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.

Détail

DOI : 10.4064/sm217-2-1

Keywords: following iterated commutators * maximal operator multilinear singular integral operators alpha multilinear fractional integral operator introduced studied eqalign * vec sup delta ldots m delta m cdots vec right alpha vec ldots m alpha m cdots vec where delta smooth truncations multilinear singular integral operators alpha multilinear fractional integral operator bmo ldots vec ldots weighted strong log type end point estimates above iterated commutators associated classes multiple weights vec vec obtained respectively