Local Hardy spaces associated with ball quasi-Banach function spaces and their dual spaces

TL;DR

This paper develops atomic decompositions for local Hardy spaces linked to ball quasi-Banach function spaces and characterizes their dual spaces, broadening understanding and applications of these function spaces.

Contribution

It establishes direct atomic decompositions for local Hardy spaces associated with ball quasi-Banach spaces and identifies their dual spaces without relying on prior relations.

Findings

Atomic decompositions for $h_X(R^n)$ are established.

Dual spaces of $h_X(R^n)$ are characterized.

Results apply to various specific ball quasi-Banach spaces.

Abstract

Let $X$ be a ball quasi-Banach function space on $\mathbb R^{n}$ and $h_{X}(\mathbb R^{n})$ the local Hardy space associated with $X$. In this paper, under some reasonable assumptions on $X$, the infinite and finite atomic decompositions for the local Hardy space $h_{X}(\mathbb R^{n})$ are established directly, without relying on the relation between $H_{X}(\mathbb R^{n})$ and $h_{X}(\mathbb R^{n})$. Moreover, we apply the finite atomic decomposition to obtain the dual space of the local Hardy space $h_{X}(\mathbb R^{n})$. Especially, the above results can be applied to several specific ball quasi-Banach function spaces, demonstrating their wide range of applications.