On weak$^*$-convergence in the localized Hardy spaces $H^1_\rho(\mathcal X)$ and its application

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 $VMO_\rho(\mathcal X)$ and show that it is the predual of the localized Hardy space $H^1_\rho(\mathcal X)$ introduced by Yang and Zhou \cite{YZ}. Then we prove a version of the classical theorem of Jones and Journ\'e \cite{JJ} on weak$^*$-convergence in $H^1_\rho(\mathcal X)$. As an application, we give an atomic characterization of $H^1_\rho(\mathcal X)$.