An RD-space X is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in X, or equivalently, that there exists a constant a0 > 1 such that for all x ∈ X and 0 < r < diam (X)=a0, the annulus B(x, a0r) n B(x, r) is nonempty, where diam (X) denotes the diameter of the metric space (X, d). An important class of RD-spaces is provided by Carnot-Caratheodory spaces with a doubling measure. In this paper, the authors introduce some spaces of Lipschitz type on RD-spaces, and discuss their relations with known Besov and Triebel-Lizorkin spaces and various Sobolev spaces.

Let (X,ρ,μ)dθbe a space of homogeneous type with d > 0 and θ∈(0,1],b be a paraaccretive function, ∈∈(0, θ]，︱s︱ < ∈, and a0 ∈(0, 1 be some constant depending on d, ∈ and s. The authors introduce the Besov space bBspq (X) with a0 < p ≤∞, and the Triebel-Lizorkin space bFspq(X) with a0 < p < ∞ and a0 < q≤∞ by first establishing a Plancherel-Polyatype inequality. Moreover, the authors establish the frame and the Littlewood-Paley function characterizations of these spaces. Furthermore, the authors introduce the new Besov space b−1 BPspq (X) and the Triebel-Lizorkin space b−1 FPspq (X). The relations among these spaces and the known Hardy-type spaces are presented. As applications, the authors also establish some real interpolation theorems, embedding theorems, Tb theorems, and the lifting property by introducing some new Riesz operators of these spaces.

Some equivalent characterizations for boundedness of maximal singular integral operators on spaces of homogeneous type are given via certain norm inequalities on John-Stromberg sharp maximal functions and without resorting the boundedness of these operators themselves. As a corollary, the results of Grafakos on Euclidean spaces are generalized to spaces of homogeneous type. Moreover, applications to maximal Monge-Ampere singular integral operators and maximal Nagel-Stein singular integral operators on certain specific smooth manifolds are also presented.

Let p ∈ (0,1] and s≧[n(1=p-1)], where [n(1=p-1)] denotes the maximal integer no more than n(1=p-1). In this paper, the authors prove that a linear operator T extends to a bounded linear operator from the Hardy space Hp(Rn) to some quasi-Banach space B if and only if T maps all (p; 2; s)-atoms into uniformly bounded elements of B.

In this paper, we introduce some new function spaces of Sobolev type on metric measure spaces. These new function spaces are defined by variants of Poincaré inequalities associated with generalized approximations of the identity, and they generalize the classical Sobolev spaces on Euclidean spaces. We then obtain two characterizations of these new Sobolev spaces including the characterization in terms of a variant of local sharp maximal functions associated with generalized approximations of the identity. For the well-known Hajłasz–Sobolev spaces on metric measure spaces, we also establish some new characterizations related to generalized approximations of the identity. Finally, we clarify the relations between the Sobolev-type spaces introduced in this paper and the Hajłasz–Sobolev spaces on metric measure spaces.