Generalized commutators of multilinear Calderon-Zygmund type operators

Let T be an m-linear Calderon-Zygmund operator with kernel K and T* be the maximal operator of T. Let S be a finite subset of Z(+) x {1,...,m} and denote d (y) over right arrow = dy(1) ... dy(m). Define the commutator T-(b) over right arrow ,T-S, of T, and T-(b) over right arrow ,T-S* of T* by T-(b) over right arrow ,T-S ((f) over right arrow)(x) = integral(Rnm) Pi((i,j)is an element of S)(b(i)(x) - b(i)(y(i))).K(x, y(1), ..., y(m))Pi(m)(j=1) f(j)(y(j))d (y) over right arrow and T-(b) over right arrow ,T-S* ((f) over right arrow)(x) = sup(delta>0) vertical bar integral(Sigma j=1m) vertical bar x-y(j)vertical bar(2) > delta(2) . Pi((i,j)is an element of S)(b(i)(x) - b(i)(y(j)))K(x, y(1), ..., y(m)) Pi(m)(j=1) f(j) (y(j))d (y) over right arrow vertical bar. These commutators are reflexible enough to generalize several kinds of commutators which already existed. We obtain the weighted strong and endpoint estimates for T-(b) over right arrow ,T-S and T-(b) over right arrow ,T-S* with multiple weights. These results are based on an estimate of the Fefferman-Stein sharp maximal function of the commutators, which is proved in a pretty much more organized way than some known proofs. Similar results for the commutators of vector-valued multilinear Calderon-Zygmund operators are also given.