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 LpLp. 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)H1(X) and BMO(X)BMO(X). In the setting of product spaces X˜=X1×⋯×Xn 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 Ap(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 H1H1 due to Mei, and Li, Pipher and Ward.