Weighted estimates for the iterated commutators of multilinear maximal and fractional type operators

The following iterated commutators T-*,(IIb) of the maximal operator for multilinear singular integral operators and I-alpha,I-IIb of the multilinear fractional integral operator are introduced and studied: T-*,T-IIb ((f) over right arrow) (x) = sub (delta) (> 0) vertical bar[b(1), [b(2), . . . [b(m-1), [b(m) , T-delta](m)](m-1) . . .](2)](1) ((f) over right arrow) (x)vertical bar, I-alpha,I- IIb ((f) over right arrow) (x) = [b(1), [b(2), . . . [b(m-1), [b(m), I-alpha](m)](m-1) . . .](2)](1) ((f) over right arrow) (x), where T-delta are the smooth truncations of the multilinear singular integral operators and I-alpha is the multilinear fractional integer operator, b(i) is an element of BMO for i =1, . . . , m and (f) over right arrow = (f(1), . . . f(m)). Weighted strong and L(log L) type end-point estimates for the above iterated commutators associated with two classes of multiple weights, A(<(p)over right) (arrow>) and A((<(p)over right) (arrow>), (q)) are obtained, respectively.