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 >= 1, 0 <= alpha < mn and <(b)over right arrow> = (b(1),..., b(m)) with each b(i) belonging to the Lipschitz space Lip(R), we denote by [(b) over right arrow, M-alpha] (resp., M-alpha,M- (b) over right arrow) the commutator of the multilinear fractional maximal operator M-alpha with (b) over right arrow (resp., the multilinear fractional maximal commutators). When alpha = 0, we denote [(b) over right arrow, M-alpha] = [(b) over right arrow, M] and M-alpha,M- (b) over right arrow = M-(b) over right arrow. We show that for 0 < s < 1, 1 < p(1),..., p(m), p, q < infinity, 1/p = 1/p(1) + ... + 1/p(m), both [(b) over right arrow, M] and M-(b) over right arrow are bounded and continuous from W-s,W- p1 (R-n) x ... x(W s,pm) (R-n) to W-s,W-p(R-n), from F-s(p1,q) (R-n) x ... x F-s (pm,q) (R-n) to F-s(p,q) (R-n) and from B-s(p1,q) (R-n) x ... x B-s (pm,q) (R-n) to B-s(p,q) (R-n). It was also shown that for 0 <= alpha < mn, 1 < p(1),..., p(m), q < infinity and 1/q = 1/p(1) + ... + 1/p(m) - alpha/n, both [<(b)over right arrow>, M] and M-(b) over right arrow are bounded from W-1,W- p1 (R-n) x ... xW(1,pm)(R-n) to W-1,W-q(R-n).