Anisotropic Hardy spaces associated with ball quasi-Banach function spaces and their applications (Q6582331)

In the paper under review, the authors introduce the Hardy space \(H^A_{X}(\mathbb{R}^n)\) associated with both \(A\) and \(X\), where \(A\) is a general expansive matrix and \(X\) is 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. \N\NThe paper contains eight sections. Section 1 is the background. In Section 2, the authors introduce the anisotropic Hardy space \(H^A_{X}(\mathbb{R}^n)\) associated with both \(A\) and \(X\) by using the nontangential grand maximal function. In Section 3, the authors characterize \(H^A_{X}(\mathbb{R}^n)\) via the radial maximal function and the nontangential maximal function. In Section 4, the authors first introduce the anisotropic \((X,q,d)\)-atom and the anisotropic atomic Hardy space \(H^{X,q,d}_{A,\ \text{atom}}(\mathbb{R}^n)\) and then state the atomic characterization of \(H^A_{X}(\mathbb{R}^n)\). The authors establish the finite atomic characterization \(H^A_{X}(\mathbb{R}^n)\) in Section 5 and characterize \(H^A_{X}(\mathbb{R}^n)\) by anisotropic molecules in Section 6. As an application, in Section 7, the authors obtain the boundedness of anisotropic Calderón-Zygmund operators \(T\) from \(H^A_{X}(\mathbb{R}^n)\) to itself or to \(X\). In Section 8, the authors apply all the main results obtained in the above sections to several specific examples of ball quasi-Banach function spaces, such as the Morrey space the Orlicz-slice space, and so on.