This paper deals with spectral assertions of Riesz potentials in some classes of quasimetric spaces. In addition we survey briefly a few related subjects: integral operators, local means and function spaces, euclidean charts of quasi–metric spaces, relations to fractal geometry.

New spaces of Lipschitz type on metric-measure spaces are introduced and they are shown to be just the well-known Besov spaces or Triebel–Lizorkin spaces when the smooth index is less than 1. These theorems also hold in the setting of spaces of homogeneous type, which include Euclidean spaces, Riemannian manifolds and some self-similar fractals. Moreover, the relationships amongst these Lipschitz-type spaces, Hajlasz–Sobolev spaces, Korevaar–Schoen–Sobolev spaces, Newtonian Sobolev space and Cheeger–Sobolev spaces on metric-measure spaces are clarified, showing that they are the same space with equivalence of norms. Furthermore, a Sobolev embedding theorem, namely that the Lipschitz-type spaces with large orders of smoothness can be embedded in Lipschitz spaces, is proved. For metric-measure spaces with heat kernels, a Hardy–Littlewood–Sobolev theorem is establish, and hence it is proved that Lipschitz-type spaces with small orders of smoothness can be embedded in Lebesgue spaces.

New norms for some distributions on spaces of homogeneous type which include some fractals are introduced. Using inhomogeneous discrete Calderon reproducing formulae and the Plancherel-Polya inequalities on spaces of homogeneous type, the authors prove that these norms give a new characterization for the Besov and Triebel-Lizorkin spaces with p; q > 1 and can be used to introduce new inhomogeneous Besov and Triebel-Lizorkin spaces with p; q≤ 1 on spaces of homogeneous type. Moreover, atomic decompositions of these new spaces are also obtained. All the results of this paper are new even for Rn.

Let (X,Э,μ)d,θ be a space of homogeneous type, i.e. X is a set, Эis a quasi-metric on X with the property that there are constants θ∈ (0，1] and C0 > 0 such that for all x; x1; y∈X, ︱Э(x,y)- Э(x1, y)︱≤C0Э(x,x1) θ[Э(x,y) +Э(x1, y)]1-Э, and μ is a nonnegative Borel regular measure on X such that for some d > 0 and all x ∈ X, μ(｛y∈X: Э(x,y)＜r｝～rd. LetЭ∈ (0,θ], ︱s︱ <ε and max｛d/(d + ε); d/(d + s + ε)｝ < q ≤∞. The author introduces new inhomogeneous Triebel-Lizorkin spaces Fs∞q(X) and establishes their frame characterizations by first establishing a Plancherel-Polya-type inequality related to the norm ‖·‖Fs∞q (X), which completes the theory of function spaces on spaces of homogeneous type. Moreover, the author establishes the connection between the space Fs∞q (X) and the homogeneous Triebel-Lizorkin space Fs∞q (X). In particular, he proves that bmo(X) coincides with F F0∞q(X).

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.