Anisotropic Hardy spaces associated with ball quasi-Banach function spaces and their applications

Abstract

Let A be a general expansive matrix and X a ball quasi-Banach function space on ${\mathbb{R}}^n$, which supports both a Fefferman–Stein vector-valued maximal inequality and the boundedness of the powered Hardy–Littlewood maximal operator on its associate space. The authors first introduce the Hardy space $H_X^A({\mathbb{R}}^n)$, associated with both A and X, via the nontangential grand maximal function, and then establish its various equivalent characterizations, respectively, in terms of radial and nontangential maximal functions, (finite) atoms, and molecules. As an application, the authors obtain the boundedness of anisotropic Calderón–Zygmund operators from $H_X^A({\mathbb{R}}^n)$ to X or to $H_X^A({\mathbb{R}}^n)$ itself via first establishing some boundedness criteria of linear operators on $H_X^A({\mathbb{R}}^n)$. All these results have a wide range of generality and, particularly, even when they are applied to the Morrey space and the Orlicz-slice space, the obtained results are also new. The novelties of this article exist in that, to overcome the essential difficulties caused by the absence of both an explicit expression and the absolute continuity of quasi-norm $\Vert \cdot {\Vert }_X$, the authors embed X into the anisotropic weighted Lebesgue space with certain special weight and then fully use the known results of the anisotropic weighted Lebesgue space.