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

TL;DR

This paper develops maximal function and Riesz transform characterizations for Hardy spaces linked to higher order elliptic operators and ball quasi-Banach spaces, broadening the understanding of these spaces in various contexts.

Contribution

It introduces new characterizations of Hardy spaces associated with elliptic operators and quasi-Banach spaces, applicable to numerous generalized Hardy spaces, including some previously unexplored cases.

Findings

Established maximal function characterizations for Hardy spaces.

Proved Riesz transform characterizations for these Hardy spaces.

Extended results to various generalized Hardy spaces such as weighted, variable, and Morrey-Hardy spaces.

Abstract

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.