Discrete Littlewood-Paley-Stein theory and multi-parameter Hardy spaces associated with flag singular integrals

TL;DR

This paper develops a unified multi-parameter Hardy space theory using discrete Littlewood-Paley-Stein analysis, avoiding traditional methods, and establishes boundedness and duality results for flag singular integral operators.

Contribution

It introduces a novel approach to multi-parameter Hardy spaces for flag singular integrals, bypassing atomic decomposition and covering lemmas, and extends the theory to all p in (0,1].

Findings

Established Hardy spaces $H^p_F$ for all $0<p extless=1$

Proved boundedness of flag singular integrals on $BMO_F$ and $H^p_F$

Developed Calderon-Zygmund decomposition and interpolation for these spaces

Abstract

The main purpose of this paper is to develop a unified approach of multi-parameter Hardy space theory using the discrete Littlewood-Paley-Stein analysis in the setting of implicit multi-parameter structure. It is motivated by the goal to establish and develop the Hardy space theory for the flag singular integral operators studied by Muller-Ricci-Stein and Nagel-Ricci-Stein. This approach enables us to avoid the use of transference method of Coifman-Weiss as often used in the $L^p$ theory for $p>1$ and establish the Hardy spaces $H^p_F$ and its dual spaces associated with the flag singular integral operators for all $0<p\leq 1$. We also prove the boundedness of flag singular integral operators on $BMO_F$ and $H^p_F$, and from $H^p_F$ to $L^p$ for all $0<p\le 1$ without using the deep atomic decomposition. As a result, it bypasses the use of Journe's type covering lemma in this implicit multi-parameter structure. The method used here provides alternate approaches of those developed by Chang, R. Fefferman, Journe and Pipher in the pure product setting. A Calderon-Zygmund decomposition and interpolation theorem are also proved for the implicit multi-parameter Hardy spaces.