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 $\Osc_{\exp L^{^τ}}(\R^{n})$ functions for $\tau \ge 1$, where $\Osc_{\exp L^{^τ}}(\R^{n})$ is a space of Orlicz type satisfying that $\Osc_{\exp L^{^τ}}(\R^{n})=\BMO(\R^{n})$ if $\tau =1$ and $\Osc_{\exp L^{^τ}}(\R^{n})\subset\BMO(\R^{n})$ 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.

In 2012, it is accepted by Bulletin of the Malaysian Mathematical Sciences Society. And there are 11 pages