Regularity of commutators of multilinear maximal operators with Lipschitz symbols

. We study the regularity properties for commutators of multilinear fractional maximal operators. More precisely, let m (cid:2) 1, 0 (cid:3) α < mn and (cid:2) b = ( b 1 ,..., b m ) with each b i belonging to the Lipschitz space Lip ( R ) , we denote by [ (cid:2) b , M α ] (resp., M α ,(cid:2) b ) the commutator of the multilinear fractional maximal operator M α with (cid:2) b (resp., the multilinear fractional maximal commutators). When α = 0, we denote [ (cid:2) b , M α ] = [ (cid:2) b , M ] and M α ,(cid:2) b = M (cid:2) b . We show that for 0 < s < 1, 1 < p 1 ,..., p m , p , q < ∞ , 1 / p = 1 / p 1 + ··· + 1 / p m , both [ (cid:2) b , M ] and M (cid:2) b are bounded and continuous from W s , p 1 ( R n ) ×···× W s , p m ( R n ) to W s , p ( R n ) , from F p 1 , q s ( R n ) × ···× F p m , q s ( R n ) to F p , q s ( R n ) and from B p 1 , q s ( R n ) ×···× B p m , q s ( R n ) to B p , q s ( R n ) . It was also shown that for 0 (cid:3) α < mn , 1 < p 1 ,..., p m , q < ∞ and 1 / q = 1 / p 1 + ··· + 1 / p m − α / n , both [ (cid:2) b , M ] and M (cid:2) b are W 1 , p 1 ( R n ) ×···× W 1 , p m ( R n ) to W 1 , q ( R n ) .