Local Hardy spaces associated with ball quasi-Banach function spaces and their dual spaces

Let X be a ball quasi-Banach function space on R^n and hₗ (R^n) the local Hardy space associated with X. In this paper, under some reasonable assumptions on X, the infinite and finite atomic decompositions for the local Hardy space hₗ (R^n) are established directly, without relying on the relation between Hₗ (R^n) and hₗ (R^n). Moreover, we apply the finite atomic decomposition to obtain the dual space of the local Hardy space hₗ (R^n). Especially, the above results can be applied to several specific ball quasi-Banach function spaces, demonstrating their wide range of applications.