Operators on Herz-type spaces associated with ball quasi-Banach function spaces

Abstract

Let $\alpha\in{\Bbb R}$, $0<p<\infty$ and $X$ be a ball quasi-Banach function space on ${\Bbb R}^n$. In this article, we introduce the Herz-type space $\dot{K}^{\alpha,p}_X({\Bbb R}^n)$ associated with $X$. We identify the dual space of $\dot{K}^{\alpha,p}_X({\Bbb R}^n)$, by which the boundedness of Hardy-Littlewood maximal operator on $\dot{K}^{\alpha,p}_X({\Bbb R}^n)$ is proved. By using the extrapolation theorem on ball quasi-Banach function spaces, we establish the extrapolation theorem on Herz-type spaces associated with ball quasi-Banach function spaces. Applying our extrapolation theorem, the boundedness of singular integral operators with rough kernels and their commutators, parametric Marcinkiewicz integrals, and oscillatory singular integral operators on $\dot{K}^{\alpha,p}_X({\Bbb R}^n)$ is obtained. As examples, we give some concrete function spaces which are members of Herz-type spaces associated with ball quasi-Banach function spaces.