In this work, we present a new approach to atoms and molecules in local Hardy spaces $h^p(\mathbb{R}^n)$ for $0<p\leq 1$, where the local vanishing moment condition is relaxed. As applications, we provide necessary and sufficient conditions for the boundedness of inhomogeneous Calderón-Zygmund operators and we prove a local version of Hardy's inequality. Based on joint works with Galia Dafni, Chun Ho Lau and Tiago Picon.