Let X be a ball quasi-Banach function space, \(\alpha \in \mathbb {R}\) and \(q\in (0,\infty )\) . In this article, the authors first introduce the Herz-type Hardy space \(\mathcal {H\dot{K}}_{X}^{\alpha ,\,q}({\mathbb {R}}^n)\) , which is defined via the non-tangential grand maximal function. Under some mild assumptions on X, the authors establish the atomic decompositions of \(\mathcal {H\dot{K}}_{X}^{\alpha ,\,q}({\mathbb {R}}^n)\) . As an application, the authors obtain the boundedness of certain sublinear operators from \(\mathcal {H\dot{K}}_{X}^{\alpha ,\,q}({\mathbb {R}}^n)\) to \(\mathcal {\dot{K}}_{X}^{\alpha ,\,q}({\mathbb {R}}^n)\) , where \(\mathcal {\dot{K}}_{X}^{\alpha ,\,q}({\mathbb {R}}^n)\) denotes the Herz-type space associated with ball quasi-Banach function space X. Finally, the authors apply these results to three concrete function spaces: Herz-type Hardy spaces with variable exponent, mixed Herz-Hardy spaces and Orlicz-Herz Hardy spaces, which belong to the family of Herz-type Hardy spaces associated with ball quasi-Banach function spaces.
