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

Let $X$ be a ball quasi-Banach function space on $\mathbb R^{n}$ and $h_{X}(\mathbb R^{n})$ 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 $h_{X}(\mathbb R^{n})$ are established directly, without relying on the relation between $H_{X}(\mathbb R^{n})$ and $h_{X}(\mathbb R^{n})$. Moreover, we apply the finite atomic decomposition to obtain the dual space of the local Hardy space $h_{X}(\mathbb R^{n})$. Especially, the above results can be applied to several specific ball quasi-Banach function spaces, demonstrating their wide range of applications.