### 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.

Original language | English |
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Pages (from-to) | 120-169 |

Number of pages | 50 |

Journal | Applied and Computational Harmonic Analysis |

Volume | 45 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jul 2018 |

### Keywords

- Spaces of homogeneous type
- Orthonormal basis
- Test function space
- Calderón reproducing formula
- Wavelet expansion
- Product Hardy space
- Carleson measure space
- BMO
- VMO
- Duality

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## Cite this

*Applied and Computational Harmonic Analysis*,

*45*(1), 120-169. https://doi.org/10.1016/j.acha.2016.09.002