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Let $\alpha\in{\Bbb R}$, $0<p<\infty$ and $X$ be a ball quasi-Banach functionspace 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 dualspace of $\dot{K}^{\alpha,p}_X({\Bbb R}^n)$, by which the boundedness ofHardy-Littlewood maximal operator on $\dot{K}^{\alpha,p}_X({\Bbb R}^n)$ isproved. By using the extrapolation theorem on ball quasi-Banach functionspaces, we establish the extrapolation theorem on Herz-type spaces associatedwith ball quasi-Banach function spaces. Applying our extrapolation theorem, theboundedness of singular integral operators with rough kernels and theircommutators, parametric Marcinkiewicz integrals, and oscillatory singularintegral operators on $\dot{K}^{\alpha,p}_X({\Bbb R}^n)$ is obtained. Asexamples, we give some concrete function spaces which are members of Herz-typespaces associated with ball quasi-Banach function spaces.

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