Flag-Like Singular Integrals and Associated Hardy Spaces on a Kind of Nilpotent Lie Groups of Step Two

The Cauchy-Szego singular integral is a fundamental tool in the study of holomorphic H-p Hardy space. But for a kind of Siegel domains, the Cauchy-Szego kernels are neither product ones nor flag ones on the Shilov boundaries, which have the structure of nilpotent Lie groups N of step two. We use the lifting method to investigate flag-like singular integrals on N, which includes these Cauchy-Szego ones as a special case. The lifting group is the product (N) over tilde of three Heisenberg groups, and naturally geometric or analytical objects on N are the projection of those on (N) over tilde. As in the flag case, we introduce various notions on N adapted to geometric feature of these kernels, such as tubes, nontangential regions, tube maximal functions, Littlewood-Paley functions, tents, shards and atoms etc. They have the feature of tri-parameters, although the second step of the group N is only 2-dimensional, i.e. there exists a hidden parameter as in the flag case. We also establish the corresponding Calderon reproducing formula, characterization of L-p(N) by Littlewood-Paley functions, L-p-boundedness of tube maximal functions and flag-like singular integrals and atomic decomposition of H-1 Hardy space on N.