Maximal Functions, Littlewood-Paley Theory, Riesz Transforms and Atomic Decomposition in the Multi-Parameter Flag Setting

Using real variable methods, Han and colleagues develop characterizations of the Hardy spaces in the multi-parameter flag setting. Novel aspects of their study are establishing appropriate discrete Calderón reproducing formulae in the flag setting and a version of the Plancherel-Pólya inequalities for flag quadratic forms, introducing the maximal function and area functions via flag Poisson kernels and a flag version of harmonic functions, and 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, they say, they obtain the full characterizations of the multi-parameter Hardy space with the flag structure.

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