Let L be a homogeneous divergence form higher order elliptic operator with complex bounded measurable coefficients on \({\mathbb {R}}^n\) and X a ball quasi-Banach function space on \({\mathbb {R}}^n\) satisfying some mild assumptions. Denote by \(H_{X,\, L}({\mathbb {R}}^n)\) the Hardy space, associated with both L and X, which is defined via the Lusin area function related to the semigroup generated by L. In this article, the authors establish both the maximal function and the Riesz transform characterizations of \(H_{X,\, L}({\mathbb {R}}^n)\) . The results obtained in this article have a wide range of generality and can be applied to the weighted Hardy space, the variable Hardy space, the mixed-norm Hardy space, the Orlicz–Hardy space, the Orlicz-slice Hardy space, and the Morrey–Hardy space, associated with L. In particular, even when L is a second order divergence form elliptic operator, both the maximal function and the Riesz transform characterizations of the mixed-norm Hardy space, the Orlicz-slice Hardy space, and the Morrey–Hardy space, associated with L, obtained in this article, are completely new.
