Haar bases on quasi-metric measure spaces, and dyadic structure theorems for function spaces on product spaces of homogeneous type

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 $\mathrm{B}\mathrm{M}\mathrm{O}(X)$. In the setting of product spaces $\tilde{X}=X_1\times \cdots \times X_n$ of homogeneous type, we show that the space $\mathrm{B}\mathrm{M}\mathrm{O}(\tilde{X})$ of functions of bounded mean oscillation on $\tilde{X}$ can be written as the intersection of finitely many dyadic $\mathrm{B}\mathrm{M}\mathrm{O}$ spaces on $\tilde{X}$, and similarly for $A_p(\tilde{X})$, reverse-H"older weights on $\tilde{X}$, and doubling weights on $\tilde{X}$. We also establish that the Hardy space $H^1(\tilde{X})$ is a sum of finitely many dyadic Hardy spaces on $\tilde{X}$, and that the strong maximal function on $\tilde{X}$ 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 $\mathrm{B}\mathrm{M}\mathrm{O}$ and $H^1$ due to Mei and to Li, Pipher and Ward.