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

摘要

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 center dot center dot center dot x 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-Holder 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. (C) 2016 Elsevier Inc. All rights reserved.