Weighted Endpoint Estimates for Multilinear Commutators of Marcinkiewicz Integrals

Let µ?,~b be the multilinear commutator generalized by µ?, the n-dimensional Marcinkiewicz integral, with Osc expL ? (R n ) functions for ? ? 1, where Osc expL ? (R n ) is a space of Orlicz type satisfying that Osc expL ? (R n ) = BMO(R n ) if ? = 1 and Osc expL ? (R n ) ? BMO(R n ) if ? > 1. The authors establish the weighted weak LlogL-type estimates for µ?,~b when ? satisfies a kind of Dini conditions.