Suppose that (X, d, µ) is a space of homogeneous type, with upper dimension

Assume that (X, d, µ) is a space of homogeneous type in the sense of Coifman and Weiss. In this article, motivated by the breakthrough work of P. Auscher and T. Hytönen on orthonormal bases of regular wavelets on spaces of homogeneous type, the authors introduce a new kind of approximations of the identity with exponential decay (for short, exp-ATI). Via such an exp-ATI, motivated by another creative idea of Y. Han et al. to merge the aforementioned orthonormal bases of regular wavelets into the frame of the existed distributional theory on spaces of homogeneous type, the authors establish the homogeneous continuous/discrete Calderón reproducing formulae on (X, d, µ), as well as their inhomogeneous counterparts. The novelty of this article exists in that d is only assumed to be a quasi-metric and the underlying measure µ a doubling measure, not necessary to satisfy the reverse doubling condition. It is well known that Calderón reproducing formulae are the cornerstone to develop analysis and, especially, harmonic analysis on spaces of homogeneous type.2010 Mathematics Subject Classification. Primary 42C40; Secondary 42B20, 42B25, 30L99. Key words and phrases. space of homogeneous type, Calderón reproducing formula, approximation of the identity, wavelet, space of test functions, distribution.

Let (𝒳, d, μ) be a space of homogeneous type, in the sense of Coifman and Weiss, with the upper dimension ω. Assume that η ∈(0, 1) is the smoothness index of the wavelets on 𝒳 constructed by Auscher and Hytönen. In this article, via grand maximal functions, the authors introduce the Hardy–Lorentz spaces H_*^{p,q}\left( \mathcal{X} \right) with the optimal range p \in \left( {{\omega \over {\omega + \eta }},\infty } \right) and q ∈ (0, ∞]. When and p \in ({\omega \over {\omega + \eta }},1]q ∈ (0, ∞], the authors establish its real-variable characterizations, respectively, in terms of radial maximal functions, non-tangential maximal functions, atoms, molecules, and various Littlewood–Paley functions. The authors also obtain its finite atomic characterization. As applications, the authors establish a real interpolation theorem on Hardy–Lorentz spaces, and also obtain the boundedness of Calderón–Zygmund operators on them including the critical cases. The novelty of this article lies in getting rid of the reverse doubling assumption of μ by fully using the geometrical properties of 𝒳 expressed via its dyadic reference points and dyadic cubes and, moreover, the results in the case q ∈ (0, 1) of this article are also new even when 𝒳 satisfies the reverse doubling condition.

Let (X,d,μ) be a space of homogeneous type, with upper dimension μ, in the sense of R. R. Coifman and G. Weiss. Let η be the Hölder regularity index of wavelets constructed by P. Auscher and T. Hytönen. In this article, the authors introduce the local Hardy space h∗,pfalse(Xfalse) via local grand maximal functions and also characterize h∗,pfalse(Xfalse) via local radial maximal functions, local non‐tangential maximal functions, local atoms and local Littlewood–Paley functions. Furthermore, the authors establish the relationship between the global and the local Hardy spaces. Finally, the authors also obtain the finite atomic characterizations of h∗,pfalse(Xfalse). As an application, the authors give the dual spaces of h∗,pfalse(Xfalse) when p∈(ω/(ω+η),1), which further completes the result of G. Dafni and H. Yue on the dual space of h∗,1false(Xfalse). This article also answers the question of R. R. Coifman and G. Weiss on the nonnecessity of any additional geometric assumptions except the doubling condition for the radial maximal function characterization of Hnormalcw1false(Xfalse) when μ(X)<∞.

In this article, the authors introduce Besov and Triebel-Lizorkin spaces on spaces of homogeneous type in the sense of Coifman and Weiss, prove that these (inhomogeneous) Besov and Triebel-Lizorkin spaces are independent of the choices of both (inhomogeneous) approximations of the identity with exponential decay and underlying spaces of distributions, and give some basic properties of these spaces. As applications, the authors show that some known function spaces coincide with certain special cases of Besov and Triebel-Lizorkin spaces and, moreover, obtain the boundedness of Calderón-Zygmund operators on these Besov and Triebel-Lizorkin spaces. All these results strongly depend on the geometrical properties, reflected via dyadic cubes, of the relevant space of homogeneous type. Compared with the known theory of these spaces on metric measure spaces, a major novelty of this article is that all results presented in this article get rid of the dependence on the reverse doubling assumption of the measure under study of the underlying space.