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In this paper, we develop via real variable methods various characterisationsof the Hardy spaces in the multi-parameter flag setting. Thesecharacterisations include those via, the non-tangential and radial maximalfunction, the Littlewood--Paley square function and area integral, Riesztransforms and the atomic decomposition in the multi-parameter flag setting.The novel ingredients in this paper include (1) establishing appropriatediscrete Calder\'on reproducing formulae in the flag setting and a version ofthe Plancherel--P\'olya inequalities for flag quadratic forms; (2) introducingthe maximal function and area function via flag Poisson kernels and flagversion of harmonic functions; (3) developing an atomic decomposition via thefinite speed propagation and area function in terms of flag heat semigroups. Asa consequence of these real variable methods, we obtain the fullcharacterisations of the multi-parameter Hardy space with the flag structure.

Maximal function, Littlewood--Paley theory, Riesz transform and atomic decomposition in the multi-parameter flag setting