The Atomic Characterization of Weighted Local Hardy Spaces and Its Applications

The purpose of this paper is to obtain atomic decomposition characterization of the weighted local Hardy space $h_{\omega }^p({\mathbb{R}}^n)$ with $\omega \in A_{\mathrm{\infty }}({\mathbb{R}}^n)$. 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 $0<p<\mathrm{\infty }$. 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.