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We provide an atomic decomposition of the product Hardy spaces$H^p(\widetilde{X})$ which were recently developed by Han, Li, and Ward in thesetting 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 homogeneoustype in the sense of Coifman and Weiss.  These Hardy spaces make use of the orthogonal wavelet bases of Auscher andHyt\"onen and their underlying reference dyadic grids.  However, no additional assumptions on the quasi-metric or on the doublingmeasure for each factor space are made. To carry out this program, we introduceproduct $(p,q)$-atoms on $\widetilde{X}$ and product atomic Hardy spaces$H^{p,q}_{{\rm at}}(\widetilde{X})$. As consequences of the atomicdecomposition of $H^p(\widetilde{X})$, we show that for all $q > 1$ the productatomic Hardy spaces coincide with the product Hardy spaces, and we show thatthe product Hardy spaces are independent of the particular choices of both thewavelet bases and the reference dyadic grids. Likewise, the product Carlesonmeasure spaces ${\rm CMO}^p(\widetilde{X})$, the bounded mean oscillation space${\rm BMO}(\widetilde{X})$, and the vanishing mean oscillation space ${\rmVMO}(\widetilde{X})$, as defined by Han, Li, and Ward, are also independent ofthe particular choices of both wavelets and reference dyadic grids.

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