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

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.