### Abstract

We give an explicit construction of Haar functions associated to a system of dyadic cubes in a geometrically doubling quasi-metric space equipped with a positive Borel measure, and show that these Haar functions form a basis for L^{p}. Next we focus on spaces X of homogeneous type in the sense of Coifman and Weiss, where we use these Haar functions to define a discrete square function, and hence to define dyadic versions of the function spaces H^{1}(X) and BMO(X). In the setting of product spaces X˜=X_{1}×⋯×X_{n} of homogeneous type, we show that the space BMO(X˜) of functions of bounded mean oscillation on X˜ can be written as the intersection of finitely many dyadic BMO spaces on X˜, and similarly for A_{p}(X˜), reverse-Hölder weights on X˜, and doubling weights on X˜. We also establish that the Hardy space H^{1}(X˜) is a sum of finitely many dyadic Hardy spaces on X˜, and that the strong maximal function on X˜ is pointwise comparable to the sum of finitely many dyadic strong maximal functions. These dyadic structure theorems generalize, to product spaces of homogeneous type, the earlier Euclidean analogues for BMO and H^{1} due to Mei, and Li, Pipher and Ward.

Original language | English |
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Pages (from-to) | 1793-1843 |

Number of pages | 51 |

Journal | Journal of Functional Analysis |

Volume | 271 |

Issue number | 7 |

DOIs | |

Publication status | Published - 1 Oct 2016 |

### Keywords

- Doubling, A, and RH weights
- Dyadic product BMO and H
- Haar bases
- Product spaces of homogeneous type

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## Cite this

*Journal of Functional Analysis*,

*271*(7), 1793-1843. https://doi.org/10.1016/j.jfa.2016.05.002