Orlicz-slice spaces.

\[main2\] Let $t\in(0,\infty)$, $q\in(1,\infty)$ and $\Phi$ be an Orlicz function with positive lower type $p_{\Phi}^-\in(1,\infty)$ and positive upper type $p_{\Phi}^+$. Then the central Hardy-Littlewood maximal function $\mathcal{M}$ is bounded on the Orlicz-slice space $(E_\Phi^q)_t(\rn)$ with the operator norm independent of $t$.

Let $t\in(0,\infty)$ and $q,\ r\in(1,\infty).$ Recall that Auscher and Prisuelos-Arribas [@ap Proposition 8.3(a)] obtained the boundedness of the Hardy-Littlewood maximal operator $\mathcal{M}$ on the space $(E_r^q)_t(\rn).$ It is easy to see that, if $\Phi(\tau):=\tau^r$ for any $\tau\in[0,\infty)$, then $(E_\Phi^q)_t(\rn)=(E_r^q)_t(\rn)$ and, in this case, Proposition \[main2\] is just [@ap Proposition 8.3(a)]. Thus, Proposition \[main2\] essentially generalizes [@ap Proposition 8.3(a)].

Let $q\in(1,\infty)$ and $\{E_k\}_{k\in\mathbb{N}}$ be a sequence of Banach spaces. The *amalgam space $\ell^q(\{E_k\}_{k\in\mathbb{N}})$* is defined to be set of all sequences $x: = \{x_k\}_{k\in\mathbb{N}}$ such that $$\|x\|_{\ell^q(\{E_k\}_{k\in\mathbb{N}})}:=\lf[\sum_{k=1}^\infty\|x_k\|_{E_k}^q\r]^{\frac{1}{q}}<\infty.$$

The following lemma comes from [@kg p.359].

\[dou\] Let $q\in(1,\infty)$. Then the space $\ell^q(\{E_k\}_{k\in\mathbb{N}})$ is a Banach space and its dual space is $\ell^{q'}(\{(E_k)^*\}_{k\in\mathbb{N}})$, where $\frac{1}{q}+\frac{1}{q'}=1$ and $(E_k)^*$ denotes the dual space of $E_k$.

\[dual\] Let $t\in(0,\infty)$, $q\in(1,\infty)$ and $\Phi$ be an Orlicz function with lower type $p_{\Phi}^-\in(1,\infty)$ and positive upper type $p_{\Phi}^+$. Let $\Psi$ be the complementary function of $\Phi$. Then the dual space of $(E_\Phi^q)_t(\rn)$ is isomorphic and homeomorphic 