Discrete Littlewood-Paley-Stein theory and multi-parameter Hardy spaces associated with flag singular integrals

The main purpose of this paper is to develop a unified approach of multi-parameter Hardy space theory using the discrete Littlewood-Paley-Stein analysis in the setting of implicit multi-parameter structure. It is motivated by the goal to establish and develop the Hardy space theory for the flag singular integral operators studied by Muller-Ricci-Stein and Nagel-Ricci-Stein. This approach enables us to avoid the use of transference method of Coifman-Weiss as often used in the $L^p$ theory for $p>1$ and establish the Hardy spaces $H^p_F$ and its dual spaces associated with the flag singular integral operators for all $0