Weighted Endpoint Estimates for Multilinear Commutators of Marcinkiewicz Integrals

Let $μ_{Ω,\vec{b}}$ be the multilinear commutator generalized by $μ_Ω$, the $n$-dimensional Marcinkiewicz integral, with $\Osc_{\exp L^{^τ}}(\R^{n})$ functions for $τ\ge 1$, 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 $τ=1$ and $\Osc_{\exp L^{^τ}}(\R^{n})\subset\BMO(\R^{n})$ if $τ>1$. The authors establish the weighted weak $L\log L$-type estimates for $μ_{Ω,\vec{b}}$ when $Ω$ satisfies a kind of Dini conditions.

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