  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 $0<p\leq 1$. We also prove the boundedness
of flag singular integral operators on $BMO_F$ and $H^p_F$, and from $H^p_F$ to
$L^p$ for all $0<p\le 1$ without using the deep atomic decomposition. As a
result, it bypasses the use of Journe's type covering lemma in this implicit
multi-parameter structure. The method used here provides alternate approaches
of those developed by Chang, R. Fefferman, Journe and Pipher in the pure
product setting. A Calderon-Zygmund decomposition and interpolation theorem are
also proved for the implicit multi-parameter Hardy spaces.
