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

Abstract

In this paper, we first show that the remarkable orthonormal wavelet expansion for L p 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 ˜ = X 1 × X 2 × ⋅ ⋅ ⋅ × X n , where each factor X i 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 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 expansion to establish the Calderón–Zygmund decomposition for product H p , 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.