On commutators of certain fractional type integrals with Lipschitz functions

In this paper, we study the commutators generated by Lipschitz functions and fractional type integral operators with kernels of the form K alpha(x,y)=kappa 1(x-A1y)kappa 2(x-A2y) ... k(m)(x-A(m)y) where 0 <= alpha = alpha(1) + ... + alpha(m) < n, each.i satisfies the (n - alpha(i))- order fractional size condition and a generalized fractional Hormander condition, A(i) is invertible, and A(i) - A(j) is invertible for i not equal j, 1 <= i, j <= m. We establish the corresponding sharp maximal function estimates and obtain the weighted Coifman type inequalities, weighted L-p(w(p)) -> L-q(w(q)) estimates, and the weighted endpoint estimates for such commutators.