Anisotropic Ball Campanato-Type Function Spaces and Their Applications

Let $A$ be a general expansive matrix and let $X$ be 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 some anisotropic ball Campanato-type function spaces associated with both $A$ and $X$, prove that these spaces are dual spaces of anisotropic Hardy spaces $H_X^A(\mathbb R^n)$ associated with both $A$ and $X$, and obtain various anisotropic Littlewood--Paley function characterizations of $H_X^A(\mathbb R^n)$. Also, as applications, the authors establish several equivalent characterizations of anisotropic ball Campanato-type function spaces, which, combined with the atomic decomposition of tent spaces associated with both $A$ and $X$, further induces their Carleson measure characterizations. All these results have a wide range of generality and, particularly, even when they are applied to Morrey spaces and Orlicz-slice spaces, some of the obtained results are also new. The novelties of this article are reflected in that, to overcome the essential difficulties caused by the absence of both an explicit expression and the absolute continuity of quasi-norm $\|\cdot\|_X$, the authors embed $X$ under consideration into the anisotropic weighted Lebesgue space with certain special weight and then fully use the known results of this weighted Lebesgue space.