Atomic decomposition of product Hardy spaces via wavelet bases on spaces of homogeneous type

TL;DR

This paper establishes an atomic decomposition for product Hardy spaces on spaces of homogeneous type using wavelet bases, demonstrating independence from specific basis choices and extending the understanding of these function spaces.

Contribution

It introduces product $(p,q)$-atoms and atomic Hardy spaces on spaces of homogeneous type, proving their equivalence to existing Hardy spaces and independence from wavelet and dyadic grid choices.

Findings

Atomic decomposition of $H^p( ilde{X})$ established

Hardy spaces shown to be basis-independent

Product Carleson measure, BMO, VMO spaces also basis-independent

Abstract

We provide an atomic decomposition of the product Hardy spaces $H^p(\widetilde{X})$ which were recently developed by Han, Li, and Ward in the setting of product spaces of homogeneous type $\widetilde{X} = X_1 \times X_2$. Here each factor $(X_i,d_i,\mu_i)$, for $i = 1$, $2$, is a space of homogeneous type in the sense of Coifman and Weiss. These Hardy spaces make use of the orthogonal wavelet bases of Auscher and Hyt\"onen and their underlying reference dyadic grids. However, no additional assumptions on the quasi-metric or on the doubling measure for each factor space are made. To carry out this program, we introduce product $(p,q)$-atoms on $\widetilde{X}$ and product atomic Hardy spaces $H^{p,q}_{{\rm at}}(\widetilde{X})$. As consequences of the atomic decomposition of $H^p(\widetilde{X})$, we show that for all $q > 1$ the product atomic Hardy spaces coincide with the product Hardy spaces, and we show that the product Hardy spaces are independent of the particular choices of both the wavelet bases and the reference dyadic grids. Likewise, the product Carleson measure spaces ${\rm CMO}^p(\widetilde{X})$, the bounded mean oscillation space ${\rm BMO}(\widetilde{X})$, and the vanishing mean oscillation space ${\rm VMO}(\widetilde{X})$, as defined by Han, Li, and Ward, are also independent of the particular choices of both wavelets and reference dyadic grids.