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

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.