Maximal function characterizations of Hardy spaces associated with both non-negative self-adjoint operators satisfying Gaussian estimates and ball quasi-Banach function spaces

Assume that L is a non-negative self-adjoint operator on L2(ℝn) with its heat kernels satisfying the so-called Gaussian upper bound estimate and that X is a ball quasi-Banach function space on ℝn satisfying some mild assumptions. Let HX, L(ℝn) be the Hardy space associated with both X and L, which is defined by the Lusin area function related to the semigroup generated by L. In this article, the authors establish various maximal function characterizations of the Hardy space HX,L(ℝn) and then apply these characterizations to obtain the solvability of the related Cauchy problem. These results have a wide range of generality and, in particular, the specific spaces X to which these results can be applied include the weighted space, the variable space, the mixed-norm space, the Orlicz space, the Orlicz-slice space, and the Morrey space. Moreover, the obtained maximal function characterizations of the mixed-norm Hardy space, the Orlicz-slice Hardy space, and the Morrey-Hardy space associated with L are completely new.