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 ℝ^n and X a ball quasi-Banach function space on ℝ^n satisfying some mild assumptions. Denote by H_X, L(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(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.