We provide an atomic decomposition of the product Hardy spaces H^p(X) which were recently developed by Han, Li, and Ward in the setting of product spaces of homogeneous type X = X₁ x 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 X and product atomic Hardy spaces H^p,q_at(X). As consequences of the atomic decomposition of H^p(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 CMO^p(X), the bounded mean oscillation space BMO(X), and the vanishing mean oscillation space VMO(X), as defined by Han, Li, and Ward, are also independent of the particular choices of both wavelets and reference dyadic grids.

The authors would like to thank the referees for careful reading and helpful comments, which made this paper more accurate. Ji Li and Lesley Ward were supported by the Australian Research Council Grant No. ARC-DP160100153.