A note on weak$^*$-convergence in $h^1(\mathbb R^d)$

TL;DR

This paper provides a simplified proof of a key result related to weak$^*$-convergence in the local Hardy space $h^1( ^d)$, clarifying foundational aspects of functional analysis in this context.

Contribution

It offers a straightforward proof of Dafni's main result on weak$^*$-convergence in $h^1( ^d)$, enhancing understanding of the space's properties.

Findings

Simplified proof of weak$^*$-convergence in $h^1( ^d)$

Clarification of foundational properties of local Hardy spaces

Improved accessibility of key functional analysis results

Abstract

In this paper, we give a very simple proof of the main result of Dafni (Canad Math Bull 45:46--59, 2002) concerning with weak$^*$-convergence in the local Hardy space $h^1(\mathbb R^d)$.