A number of recent results in Euclidean harmonic analysis have exploited several adjacent systems of dyadic cubes, instead of just one fixed system. In this paper, we extend such constructions to general spaces of homogeneous type, making these tools available for analysis on metric spaces. The results include a new (non-random) construction of boundedly many adjacent dyadic systems with useful covering properties, and a streamlined version of the random construction recently devised by H. Martikainen and the first author. We illustrate the usefulness of these constructions with applications to weighted inequalities and the BMO space; further applications will appear in forthcoming work.

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 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 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 to Li, Pipher and Ward.

We characterize two-weight norm inequalities for potential type integral operators in terms of Sawyer-type testing conditions. Our result is stated in a space of homogeneous type with no additional geometric assumptions, such as group structure or non-empty annulus property, which appeared in earlier works on the subject. One of the new ingredients in the proof is the use of a finite collection of adjacent dyadic systems recently constructed by the author and T. Hytönen. We further extend the previous Euclidean characterization of two-weight norm inequalities for fractional maximal functions into spaces of homogeneous type.2010 Mathematics Subject Classification. 42B25 (30L99, 47B38).

Abstract. We give an intrinsic characterization of all subsets of a doubling metric space that can arise as a member of some system of dyadic cubes on the underlying space, as constructed by Christ.

We consider a version of M. Riesz fractional integral operator on a space of homogeneous type and show an analogue of the well-known Hardy-Littlewood-Sobolev theorem in this context. In our main result, we investigate the dependence of the operator norm on weighted spaces on the weight constant, and find the relationship between these two quantities. It it shown that the estimate obtained is sharp in any given space of homogeneous type with infinitely many points. Our result generalizes the recent Euclidean result by Lacey, Moen, Pérez and Torres [21]. q } Ap,q , and the estimate is sharp. The main result in the present paper is the extension of this estimate into general spaces of homogeneous type.2010 Mathematics Subject Classification. 42B25 (30L99, 47B38). Key words and phrases. Fractional integral, a space of homogeneous type, weighted norm inequalities, sharp bounds.