The Atomic Characterization of Weighted Local Hardy Spaces and Its Applications

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 Calderón'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 Calderón--Zygmund singular integrals and local fractional integrals on weighted local Hardy spaces.

30 pages