Herz-Type Hardy Spaces Associated with Ball Quasi-Banach Function Spaces

Abstract

Let $X$ be a ball quasi-Banach function space, $\hat{I}\pm \in \mathbb{R}$ and $q\in (0,\infty )$. In this paper, the authors first introduce the Herz-type Hardy space ${\mathcal{H}\dot{\mathcal{K}}}_X^{\hat{I}\pm ,q}({\mathbb{R}}^n)$, which is defined via the non-tangential grand maximal function. Under some mild assumptions on $X$, the authors establish the atomic decompositions of ${\mathcal{H}\dot{\mathcal{K}}}_X^{\hat{I}\pm ,q}({\mathbb{R}}^n)$. As an application, the authors obtain the boundedness of certain sublinear operators from ${\mathcal{H}\dot{\mathcal{K}}}_X^{\hat{I}\pm ,q}({\mathbb{R}}^n)$ to ${\dot{\mathcal{K}}}_X^{\hat{I}\pm ,q}({\mathbb{R}}^n)$, where ${\dot{\mathcal{K}}}_X^{\hat{I}\pm ,q}({\mathbb{R}}^n)$ denotes the Herz-type space associated with ball quasi-Banach function space $X$. Finally, the authors apply these results to three concrete function spaces: Herz-type Hardy spaces with variable exponent, mixed Herz-Hardy spaces and Orlicz-Herz Hardy spaces, which belong to the family of Herz-type Hardy spaces associated with ball quasi-Banach function spaces.

27 pages, submitted