Maximal Function and Riesz Transform Characterizations of Hardy Spaces Associated with Homogeneous Higher Order Elliptic Operators and Ball Quasi-Banach Function Spaces

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 totally new.