On weak${}^{\ast }$-convergence in the localized Hardy spaces $H_{\rho }^1(\mathcal{X})$ 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 $x,y\in \mathcal{X}$, $$\rho (y)\le C_0[\rho (x){]}^{1/(1+k_0)}[\rho (x)+d(x,y){]}^{k_0/(1+k_0)}.$$ In this paper, we define a space $VM{O}_{\rho }(\mathcal{X})$ and show that it is the predual of the localized Hardy space $H_{\rho }^1(\mathcal{X})$ introduced by Yang and Zhou \cite{YZ}. Then we prove a version of the classical theorem of Jones and Journé \cite{JJ} on weak${}^{\ast }$-convergence in $H_{\rho }^1(\mathcal{X})$. As an application, we give an atomic characterization of $H_{\rho }^1(\mathcal{X})$.