Herz-slice spaces and applications

Let $\alpha \in {\mathbb{R}}^n$, $t\in (0,\mathrm{\infty })$, $p\in (0,\mathrm{\infty }]$, $r\in (1,\mathrm{\infty })$ and $q\in [1,\mathrm{\infty }]$. We introduce the homogeneous Herz-slice space $(\dot{K}{E}_{q,r}^{\alpha ,p}{)}_t({\mathbb{R}}^n)$, the non-homogeneous Herz-slice space $(K{E}_{q,r}^{\alpha ,p}{)}_t({\mathbb{R}}^n)$ and show some properties of them. As an application, the bounds for the Hardy–Littlewood maximal operator on these spaces is considered.