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In this paper, using the remarkable orthonormal wavelet basis constructedrecently by Auscher and Hyt\"onen, we establish the theory of product Hardyspaces on spaces ${\widetilde X} = X_1\times X_2\times\cdot \cdot\cdot\timesX_n$, where each factor $X_i$ is a space of homogeneous type in the sense ofCoifman and Weiss. The main tool we develop is the Littlewood--Paley theory on$\widetilde X$, which in turn is a consequence of a corresponding theory oneach factor space. We define the square function for this theory in terms ofthe wavelet coefficients. The Hardy space theory developed in this paperincludes product~$H^p$, the dual $\cmo^p$ of $H^p$ with the special case $\bmo= \cmo^1$, and the predual $\vmo$ of $H^1$. We also use the wavelet expansionto establish the Calder\'on--Zygmund decomposition for product $H^p$, anddeduce an interpolation theorem. We make no additional assumptions on thequasi-metric or the doubling measure for each factor space, and thus we extendto the full generality of product spaces of homogeneous type the aspects ofboth one-parameter and multiparameter theory involving the Littlewood--Paleytheory and function spaces. Moreover, our methods would be expected to be apowerful tool for developing wavelet analysis on spaces of homogeneous type.

Hardy space theory on spaces of homogeneous type via orthonormal wavelet bases