Endpoint estimates for commutators of mutilinear square function satisfying some integrable condition

In this paper, the (Lp1 × · · · × Lpm , Lq)-estimate for the commutator TΠb generalized by multilinear square function T and Lipschitz function b is established for 1/q = ∑m 1 − β , pi p0≥ 1. Meanwhile, we also establish (Lp1 × · · · × Lpm , Λ˙ β− n )-q n n j=1 pi n p boundedness and (L β1 × · · · × L βm , BMO)-estimates for the commutator TΠb. Finally, the (Lp1 × · · · × Lpm , F˙ β, )-boundedness is obtained, too.