Compactness of the commutator of multilinear fourier multiplier operator on weighted lebesgue space

Let t σ be the multilinear fourier multiplier operator associated with multiplier σ satisfying the sobolev regularity that s u p l z σ l w s 1,.,s m (r m n) < ∞ for some s k (n / 2,n ] (k = 1,.,m). the authors prove that if b 1,.,b m b m o (r n) and w ∏ k = 1 m a p k / t k (t k = n / s k),then the commutator t σ,σ b is bounded from l p 1 (r n,w 1) × × l p m (r n,w m) to l p (r n,w ). moreover,the authors also prove that if b 1,.,b m v m o (r n) and w ∏ k = 1 m a p k / t k (t k = n / s k),then the commutator t σ,σ b is compact operator from l p 1 (r n,w 1) × ⋯ × l p m (r n,w m) to l p (r n,w ). © 2014 jiang zhou and peng li.