type $p_\Phi^-=1$ is also a Young function.

2.  Let $q\in[1,\infty)$ and $\Phi$ be a Young function with lower type $p_{\Phi}^-\in[1,\infty)$ and positive upper type $p_{\Phi}^+$. We know that $L^\Phi(\rn)$ is a Banach space (see [@mmz p.67, Theorem 10]). Then it is easy to prove that $(E_\Phi^q)_t(\rn)$ is also a Banach space.

The following two lemmas come from [@mmz p.13, Proposition 1(ii); p.58, Proposition 1], respectively.

\[un\] Let $\Phi$ be an N-function and $\Psi$ its complementary function. Then $\Phi$ and $\Psi$ are strictly increasing and hence their inverses $\Phi^{-1}$ and $\Psi^{-1}$ are uniquely defined and, for any $t\in(0,\infty)$, $$t<\Phi^{-1}(t)\Psi^{-1}(t)<2t.$$

\[hold\] Let $\Phi$ be a Young function and $\Psi$ its complementary function. If $f\in L^\Phi(\rn)$ and $g\in L^\Psi(\rn)$, then $$\int_{\rn}\lf|f(x)g(x)\r|dx\le2\|f\|_{L^\Phi(\rn)}\|g\|_{L^\Psi(\rn)}.$$

The following Fefferman-Stein type inequality for Orlicz-slice spaces extends the well-known Fefferman-Stein vector-valued maximal inequality [@fs Theorem 1(1)] , which plays an important role in the succeeding sections.

\[main\] Let $t\in(0,\infty)$, $q,\ r\in(1,\infty)$ and $\Phi$ be an Orlicz function with lower type $p_{\Phi}^-\in(1,\infty)$ and positive upper type $p_{\Phi}^+$. Then there exists a positive constant $C$ such that, for any $\{f_j\}_{j\in\mathbb{Z}}\subset (E_\Phi^q)_t(\rn)$, $$\left\|\left\{\sum _{j\in\mathbb{Z}}\left[\mathcal{M}(f_j)\right]^r\right\}
^{\frac{1}{r}}\right\|_{(E_\Phi^q)_t(\rn)}
\le C\left\|\left\{\sum _{j\in\mathbb{Z}}|f_j|^r\right\}^{\frac{1}{r}}\right\|_{(E_\Phi^q)_t(\rn)},$$ where $C$ is independent of $\{f_j\}_{j\in\mathbb{Z}}$ and $t$.

For any $\{f_j\}_{j\in\mathbb{Z}}\subset (E_\Phi^q)_t(\rn)$, applying Lemma \[mm\], 