Abstract

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 a ₀ > 1 such that for all x ∈ 𝒳 and 0 < r < diam(𝒳)/a ₀, the annulus B(x, a ₀ r) \ 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-Carathéodory 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. As an application, a difference characterization of Besov and Triebel-Lizorkin spaces on RD-spaces is obtained.