  We introduce a new class of weighted local approximate atoms including
classical weighted local atoms. Then we further obtain the weighted local
approximate atomic decompositions of weighted local Hardy spaces $h_{\omega}
^p(R^n)$ with $0<p\leq 1$ and weight $\omega\in A_1(R^n)$. As an application,
we prove the boundedness of inhomogeneous Calder\'on-Zygmund operators on
$h_{\omega}^p(R^n)$ via weighted local approximate atoms and molecules.
