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

We provide an atomic decomposition of the product Hardy spaces H^p ({formula presented}) which were recently developed by Han, Li, and Ward in the setting of product spaces of homogeneous type {formula presented} = X₁ × X₂. Here each factor (X_i, d_i, μ_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 Hytönen 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 {formula presented} and product atomic Hardy spaces {formula presented}. As consequences of the atomic decomposition of {formula presented}, 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 CMO^p({formula presented}), the bounded mean oscillation space BMO({formula presented}), and the vanishing mean oscillation space VMO({formula presented}), as defined by Han, Li, and Ward, are also independent of the particular choices of both wavelets and reference dyadic grids.