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

TL;DR

This paper introduces and analyzes weighted estimates for iterated commutators of multilinear maximal and fractional integral operators, providing new bounds and endpoint estimates in the context of multiple weights.

Contribution

It develops weighted strong and endpoint estimates for new classes of iterated commutators of multilinear operators, extending previous results in harmonic analysis.

Findings

Established weighted strong type estimates for the commutators.

Derived $L( ext{log}L)$ endpoint estimates for the operators.

Extended the theory to multiple weight classes $A_{oldsymbol{p}}$ and $A_{(oldsymbol{p}, q)}$.

Abstract

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.