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 ofdyadic cubes in a geometrically doubling quasi-metric space equipped with apositive 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 Coifmanand Weiss, where we use these Haar functions to define a discrete squarefunction, 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 ${\rmBMO}(\widetilde{X})$ 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 $H^1(\widetilde{X})$ isa sum of finitely many dyadic Hardy spaces on $\widetilde{X}$, and that thestrong maximal function on $\widetilde{X}$ is pointwise comparable to the sumof finitely many dyadic strong maximal functions. These dyadic structuretheorems generalize, to product spaces of homogeneous type, the earlierEuclidean analogues for ${\rm BMO}$ and $H^1$ due to Mei and to Li, Pipher andWard.