Product Hardy spaces meet ball quasi-Banach function spaces

TL;DR

This paper develops a comprehensive theory of product Hardy spaces based on ball quasi-Banach function spaces, establishing decomposition, extrapolation, and boundedness results for two-parameter singular integral operators.

Contribution

It introduces new product Hardy spaces associated with ball quasi-Banach function spaces and proves their decomposition and boundedness properties, extending classical harmonic analysis results.

Findings

Decomposition theorem for ${H}_X( imes)$ spaces using Calderón's identity

Extrapolation theorems for Rubio de Francia on $X( imes)$ spaces

Boundedness of two-parameter singular integrals on these spaces

Abstract

The main purpose of this paper is to develop the theory of product Hardy spaces built on Banach lattices on $\mathbb R^n\times\mathbb R^m$. First we introduce new product Hardy spaces ${H}_X(\mathbb R^n\times\mathbb R^m)$ associated with ball quasi-Banach function spaces $X(\mathbb R^n\times\mathbb R^m)$ via applying the Littlewood-Paley-Stein theory. Then we establish a decomposition theorem for ${H}_X(\mathbb R^n\times\mathbb R^m)$ in terms of the discrete Calder\'on's identity. Moreover, we explore some useful and general extrapolation theorems of Rubio de Francia on $X(\mathbb R^n\times\mathbb R^m)$ and give some applications to boundedness of operators. Finally, we conclude that the two-parameter singular integral operators $\widetilde T$ are bounded from ${H}_X(\mathbb R^n\times\mathbb R^m)$ to itself and bounded from ${H}_X(\mathbb R^n\times\mathbb R^m)$ to $X(\mathbb R^n\times\mathbb R^m)$ via extrapolation. The main results obtained in this paper have a wide range of generality. Especially, we can apply these results to many concrete examples of ball quasi-Banach function spaces, including product Herz spaces, weighted product Morrey spaces and product Musielak--Orlicz--type spaces.