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