Commutators of integral operators with variable kernels on Hardy spaces

Let T-Omega.alpha (0 <= alpha < n) be the singular and fractional integrals with variable kernel Omega ( x, z), and [b. T-Omega.alpha] be the commutator generated by T-Omega.alpha and a Lipschitz function b. In this paper, the authors study the boundedness of [b, T-Omega.alpha] on the Hardy spaces, under some assumptions such as the L-r-Dini condition. Similar results and the weak type estimates at the end-point cases are also given for the homogeneous convolution Operators T-Omega.alpha (0 <= alpha < n). The smoothness conditions imposed on Omega are weaker than the corresponding known results.