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

The Atomic Characterization of Weighted Local Hardy Spaces and Its Applications