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

In this paper, we first show that the remarkable orthonormal wavelet expansion for Lp constructed recently by Auscher and Hytönen also converges in certain spaces of test functions and distributions. Hence we establish the theory of product Hardy spaces on spaces X˜=X1×X2×⋅⋅⋅×Xn, where each factor Xi is a space of homogeneous type in the sense of Coifman and Weiss. The main tool we develop is the Littlewood-Paley theory on X˜, which in turn is a consequence of a corresponding theory on each factor space. We define the square function for this theory in terms of the wavelet coefficients. The Hardy space theory developed in this paper includes product Hp, the dual CMOp of Hp with the special case BMO=CMO1, and the predual VMO of H1. We also use the wavelet expansion to establish the Calderón-Zygmund decomposition for product Hp, and deduce an interpolation theorem. We make no additional assumptions on the quasi-metric or the doubling measure for each factor space, and thus we extend to the full generality of product spaces of homogeneous type the aspects of both one-parameter and multiparameter theory involving the Littlewood-Paley theory and function spaces. Moreover, our methods would be expected to be a powerful tool for developing wavelet analysis on spaces of homogeneous type.