Weighted Endpoint Estimates for Multilinear Commutators of Marcinkiewicz Integrals

Abstract: Let $\mu_{\Omega,\vec{b}}$ be the multilinear commutator generalized by $\mu_{\Omega}$, the $n$-dimensional Marcinkiewicz integral, with $\Osc_{\exp L^{^{\tau}}}(\R^{n})$ functions for $\tau\ge 1$, where $\Osc_{\exp L^{^{\tau}}}(\R^{n})$ is a space of Orlicz type satisfying that $\Osc_{\exp L^{^{\tau}}}(\R^{n})=\BMO(\R^{n})$ if $\tau=1$ and $\Osc_{\exp L^{^{\tau}}}(\R^{n})\subset\BMO(\R^{n})$ if $\tau>1$. The authors establish the weighted weak $L\log L$-type estimates for $\mu_{\Omega,\vec{b}}$ when $\Omega$ satisfies a kind of Dini conditions.