  In this paper, we develop via real variable methods various characterisations
of the Hardy spaces in the multi-parameter flag setting. These
characterisations include those via, the non-tangential and radial maximal
function, the Littlewood--Paley square function and area integral, Riesz
transforms and the atomic decomposition in the multi-parameter flag setting.
The novel ingredients in this paper include (1) establishing appropriate
discrete Calder\'on reproducing formulae in the flag setting and a version of
the Plancherel--P\'olya inequalities for flag quadratic forms; (2) introducing
the maximal function and area function via flag Poisson kernels and flag
version of harmonic functions; (3) developing an atomic decomposition via the
finite speed propagation and area function in terms of flag heat semigroups. As
a consequence of these real variable methods, we obtain the full
characterisations of the multi-parameter Hardy space with the flag structure.

Classical Analysis and ODEs;Complex Variables