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 H¹(X) and BMO(X). In the setting of product spaces X˜=X₁×⋯×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 H¹(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 H¹ due to Mei, and Li, Pipher and Ward.