There are two folds of this article. The first part is concentrating on estimates for generalized Calderón-Zygmund operators acting on Hardy spaces H p (G). Here G is a simply connected homogeneous Lie group. We also obtained estimates on the spaces L ∞ (G) and BMO p (G). The second part of this article is applications of results from the first part to the (Formula presented.) -Neumann problem on bounded, smoothly pseudoconvex domains in C n+1 . We obtain H p estimates for the Calderón operator when G = H n , the n-dimensional Heisenberg group.