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

Let X be a ball quasi-Banach function space on Rⁿ and hX( Rⁿ) 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 hX( Rⁿ) are established directly, without relying on the relation between HX( Rⁿ) and hX( Rⁿ). Moreover, we apply the finite atomic decomposition to obtain the dual space of the local Hardy space hX( Rⁿ). Especially, the above results can be applied to several specific ball quasi-Banach function spaces, demonstrating their wide range of applications.