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

TL;DR

This paper constructs Haar bases on quasi-metric spaces with measures, and establishes dyadic structure theorems for function spaces like BMO and H^1 on product spaces of homogeneous type, generalizing Euclidean results.

Contribution

It provides explicit Haar bases in quasi-metric spaces and proves dyadic decompositions for BMO, H^1, and related spaces on product spaces of homogeneous type.

Findings

Haar functions form a basis for L^p in quasi-metric spaces.

BMO and A_p spaces can be expressed as intersections of dyadic spaces.

H^1 and maximal functions decompose into sums of dyadic counterparts.

Abstract

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.