Characterizations of product Hardy spaces on stratified groups by singular integrals and maximal functions

TL;DR

This paper extends the theory of Hardy spaces on stratified Lie groups by providing characterizations via singular integrals and maximal functions, filling important gaps in the classical harmonic analysis framework.

Contribution

It introduces new techniques to characterize Hardy spaces on stratified groups through singular integrals and maximal functions, building on recent advances on the Heisenberg group.

Findings

Hardy spaces characterized by singular integrals on stratified groups

Hardy spaces characterized by maximal functions on stratified groups

New techniques developed for product of stratified groups

Abstract

A large part of the theory of Hardy spaces on products of Euclidean spaces has been extended to the setting of products of stratified Lie groups. This includes characterisation of Hardy spaces by square functions and by atomic decompositions, proof of the duality of Hardy spaces with BMO, and description of many interpolation spaces. Until now, however, two aspects of the classical theory have been conspicuously absent: the characterisation of Hardy spaces by singular integrals (of Christ--Geller type) or by (vertical or nontangential) maximal functions. In this paper we fill in these gaps by developing new techniques on products of stratified groups, using the ideas of Chen, Cowling, Lee, Li and Ottazzi on the Heisenberg group with flag structure.