Some recent works on multi-parameter Hardy space theory and discrete Littlewood-Paley Analysis

The main purpose of this paper is to briefly review the earlier works of multiparameter Hardy space theory and boundedness of singular integral operators on such spaces defined on product of Euclidean spaces, and to describe some recent developments in this direction. These recent works include discrete multiparameter Calderón reproducing formulas and Littlewood-Paley theory in the framework of product of two homogeneous spaces, product of Carnot-Caretheodory spaces, multiparameter structures associated with flag singular integrals and the Zygmund dilation. Using these discrete multiparameter analysis, we are able to establish the theory of multiparameter Hardy spaces associated to the aforementioned multiparameter structures and prove the boundedness of singular integral operators on such Hardy H spaces and from H to L for all 0 < p ≤ 1, and derive the dual spaces of the Hardy spaces. These Hardy spaces are canonical and intrinsic to the underlying structures since they satisfy Calderón-Zygmund decomposition for functions in such spaces and interpolation properties between them. Proving boundedness of singular integral operators on product Hardy spaces was an extremely difficult task two decades ago. Our method avoids the use of very difficult Journe’s geometric lemma and is a unified approach to the multiparameter theory of Hardy spaces in all aforementioned settings.