Weighted Estimates for Multilinear Commutators of Marcinkiewicz Integrals with Bounded Kernel

Let $\mu_{\Omega,\vec{b}}$ be the multilinear commutator generalized by $\mu_{\Omega}$, the $n$-dimensional Marcinkiewicz integral with the bounded kernel, and $b_{j}\in \Osc_{\exp L^{r_{j}}}(1\le j\le m)$. In this paper, the following weighted inequalities are proved for $\omega\in A_{\infty}$ and $0<p<\infty$, $$\|\mu_{\Omega}(f)\|_{L^{p}(\omega)}\leq C\|M(f)\|_{L^{p}(\omega)}, \ \ \|\mu_{\Omega,\vec{b}}(f)\|_{L^{p}(\omega)}\leq C\|M_{L(\log L)^{1/r}}(f)\|_{L^{p}(\omega)}.$$ The weighted weak $L(\log L)^{1/r}$ -type estimate is also established when $p=1$ and $\omega\in A_{1}$.