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

TL;DR

This paper develops comprehensive real-variable characterizations of Hardy spaces in the multi-parameter flag setting, including maximal functions, Littlewood--Paley theory, Riesz transforms, and atomic decompositions, advancing harmonic analysis techniques.

Contribution

It introduces novel discrete Calderón reproducing formulae, flag-specific maximal and area functions, and an atomic decomposition framework for multi-parameter Hardy spaces.

Findings

Full characterizations of multi-parameter Hardy spaces with flag structure.

New Calderón reproducing formulae and Plancherel--Pólya inequalities for flag settings.

Atomic decomposition via flag heat semigroups and finite speed propagation.

Abstract

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.