Product Hardy Spaces on Spaces of Homogeneous Type: Discrete Product Calder\'on-Type Reproducing Formula, Atomic Characterization, and Product Calder\'on--Zygmund Operators

TL;DR

This paper develops a new atomic characterization of product Hardy spaces on spaces of homogeneous type using a discrete Calderón reproducing formula, enabling analysis of boundedness of operators like Calderón--Zygmund operators.

Contribution

It introduces a novel discrete Calderón-type reproducing formula with bounded support, facilitating atomic characterization and operator boundedness in product Hardy spaces.

Findings

Established atomic characterization of product Hardy spaces.

Proved boundedness of product Calderón--Zygmund operators on these spaces.

Provided a criterion for operator boundedness from Hardy to Lebesgue spaces.

Abstract

Let $i\in\{1,2\}$ and $X_i$ be a space of homogeneous type in the sense of Coifman and Weiss with the upper dimension $\omega_i$. Also let $\eta_i$ be the smoothness index of the Auscher--Hyt\"onen wavelet function $\psi^{k_i}_{\alpha_i}$ on $X_i$. In this article, for any $p\in(\max\{\frac{\omega_1}{\omega_1+\eta_1},\frac{\omega_2}{\omega_2+\eta_2}\}, 1]$, by regarding the product Carleson measure space $\mathrm{CMO}^p_{L^2}(X_1\times X_2)$ as the test function space and its dual space $(\mathrm{CMO}^p_{L^2}(X_1\times X_2))'$ as the corresponding distribution space, we introduce the product Hardy space $H^p(X_1\times X_2)$ in terms of wavelet coefficients. Moreover, we establish an atomic characterization of this product Hardy space and, as an application, obtain a criterion for the boundedness of linear operators from product Hardy spaces to corresponding Lebesgue spaces. To escape the wavelet reproducing formula, which is not useful for this atomic characterization because the wavelets have no bounded support, we establish a new discrete product Calder\'on-type reproducing formula, which holds in the product Hardy space and has bounded support. This reproducing formula also leads to the boundedness of product Calder\'on--Zygmund operators on the product Hardy space.