Journal of the
Korean Mathematical Society JKMS

In this paper, we investigate the weighted vector-valued \linebreak bounds for a class of multilinear singular integral operators, and its commutators, from $L^{p_1}(l^{q_1};\,\mathbb{R}^n,w_1)\times\dots\times L^{p_m}(l^{q_m};\,\mathbb{R}^n,w_m)$ to $L^{p}(l^q;\mathbb{R}^n$, $\nu_{\vec{w}})$, with $p_1,\ldots,p_m$, $q_1,\ldots,q_m\in (1,\,\infty)$, $1/p=1/p_1+\dots+1/p_m$, $1/q=1/q_1+\dots+1/q_m$ and $\vec{w}=(w_1,\ldots,w_m)$ a multiple $A_{\vec{P}}$ weights. Our argument also leads to the weighted weak type endpoint estimates for the commutators. As applications, we obtain some new weighted estimates for the Calder\'on commutator.

Keywords: weighted vector-valued inequality, multilinear singular integral operator, commutator, non-smooth kernel, multiple weight

MSC numbers: 42B20