On weak*-convergence in the localized Hardy spaces H¹ρ(X) and its application

الأصل · EN

Let (X, d, μ) be a complete RD-space. Let ρ be an admissible function on X, which means that ρ is a positive function on X and there exist positive constants C₀ and k₀ such that, for any x,y∈ X, ρ(y)≤ C₀ [ρ(x)]¹/⁽¹⁺ᵏ⁰⁾ [ρ(x)+d(x,y)]ᵏ⁰/⁽¹⁺ᵏ⁰⁾. In this paper, we define a space VMOρ(X) and show that it is the predual of the localized Hardy space H¹ρ(X) introduced by Yang and Zhou YZ. Then we prove a version of the classical theorem of Jones and Journé JJ on weak*-convergence in H¹ρ(X). As an application, we give an atomic characterization of H¹ρ(X).