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

TL;DR

This paper studies weak$^*$-convergence in localized Hardy spaces on RD-spaces, introduces a new VMO space as its predual, and provides an atomic characterization of these spaces, extending classical harmonic analysis results.

Contribution

It defines a new VMO space as the predual of localized Hardy spaces and establishes weak$^*$-convergence results in this setting, extending classical theorems.

Findings

Established $VMO_ ho( ext{X})$ as the predual of $H^1_ ho( ext{X})$.

Proved a weak$^*$-convergence theorem in $H^1_ ho( ext{X})$.

Provided an atomic characterization of $H^1_ ho( ext{X})$.

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 $VMO_\rho(\mathcal X)$ 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 $H^1_\rho(\mathcal X)$.