Flag-like singular integrals and associated Hardy spaces on a kind of nilpotent Lie groups of step two

This is the authors' abstract. We don't add key points for this paper.

The Cauchy-Szeg\"o singular integral is a fundamental tool in the study of holomorphic Hᵖ Hardy space. But for a kind of Siegel domains, the Cauchy-Szeg\"o 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-Szeg\"o ones as a special case. The lifting group is the product N ̃ of three Heisenberg groups, and naturally geometric or analytical objects on N are the projection of those on N ̃. 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 Calder\'on 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 H¹ Hardy space on N.