On Weak$^*$-convergence in the Localized Hardy Spaces $H^1_\rho(\mathcal{X})$ and its Application

Abstract

Let $(\mathcal{X}, d, \mu)$ be a complete RD-space. Let $\rho$ be an admissible function on $\mathcal{X}$, which means that $\rho$ is a positive function on $\mathcal{X}$ and there exist positive constants $C_0$ and $k_0$ such that, for any $x, y \in \mathcal{X}$,\[ \rho(y) \leq C_0 [\rho(x)]^{1/(1+k_0)} [\rho(x) + d(x, y)]^{k_0/(1+k_0)}.\]In this paper, we define a space $\operatorname{VMO}_\rho(\mathcal{X})$ and show that it is the predual of the localized Hardy space $H^1_\rho(\mathcal{X})$ introduced by Yang and Zhou [14]. Then we prove a version of the classical theorem of Jones and Journé [7] on weak$^*$-convergence in $H^1_\rho(\mathcal{X})$. As an application, we give an atomic characterization of $H^1_\rho(\mathcal{X})$.