On weak$^*$-convergence in the Hardy space $H^1$ over spaces of homogeneous type

TL;DR

This paper proves that weak* convergence holds in the Hardy space H^1 over complete spaces of homogeneous type, extending the understanding of convergence properties in these function spaces.

Contribution

It establishes the weak* convergence property in H^1 spaces over spaces of homogeneous type, a result previously not confirmed in this general setting.

Findings

Weak* convergence holds in H^1( ext{X}) for spaces of homogeneous type.

Extension of weak* convergence results to more general metric measure spaces.

Provides foundational support for analysis in Hardy spaces over complex spaces.

Abstract

Let $\mathcal X$ be a complete space of homogeneous type. In this note, we prove that the weak$^*$-convergence is true in the Hardy space $H^1(\mathcal X)$ of Coifman and Weiss.