The Atomic Characterization of Weighted Local Hardy Spaces and Its Applications

TL;DR

This paper establishes an atomic decomposition for weighted local Hardy spaces using Calderón's identity and Littlewood--Paley theory, improving previous results and applying them to boundedness of certain integral operators.

Contribution

It provides a new atomic characterization of weighted local Hardy spaces, enhancing existing decomposition theorems and broadening their applications.

Findings

Atomic decomposition characterization established.

Improved previous atomic decomposition results.

Boundedness of Calderón--Zygmund and fractional integrals demonstrated.

Abstract

The purpose of this paper is to obtain atomic decomposition characterization of the weighted local Hardy space $h_{\omega}^{p}(\mathbb {R}^{n})$ with $\omega\in A_{\infty}(\mathbb {R}^{n})$. We apply the discrete version of Calder\'on's identity and the weighted Littlewood--Paley--Stein theory to prove that $h_{\omega}^{p}(\mathbb {R}^{n})$ coincides with the weighted$\text{-}(p,q,s)$ atomic local Hardy space $h_{\omega,atom}^{p,q,s}(\mathbb {R}^{n})$ for $0<p<\infty$. The atomic decomposition theorems in our paper improve the previous atomic decomposition results of local weighted Hardy spaces in the literature. As applications, we derive the boundedness of inhomogeneous Calder\'on--Zygmund singular integrals and local fractional integrals on weighted local Hardy spaces.