follows, by Lemma \[lem2\], without loss of generality, we may *always assume* that an Orlicz function $\Phi$ is continuous and strictly increasing.

\[lem3\] Let $\Phi$ be an Orlicz function with positive lower type $p_{\Phi}^-$. If the inequality that $$\int_{\rn}\Phi\lf(\frac{|f(x)|}{\lambda}\r)\,dx\leqslant\widetilde{C}\quad for\ some\ \lambda\in(0,\infty)
\ and\ positive\ constant\ \widetilde{C}$$ holds true, then there exists a positive constant $C$, depending on $\widetilde{C}$ and $p_{\Phi}^-$, such that $\|f\|_{L^\Phi(\rn)}\le C\lambda$.

The proof is simple and we can take $C:=(1+\widetilde{C}C_{(p_{\Phi}^-)})^{1/p_{\Phi}^-}$ with $\widetilde{C}$ as in the assumption of Lemma \[lem3\]. This finishes the proof of Lemma \[lem3\].

Now we introduce the Orlicz-slice space and the Orlicz-amalgam space. The former is a generalization of the slice spaces introduced in [@am; @ap], and the latter is a generalization of the classical amalgam space $(L^p,\ell^q)$ defined by N. Wiener in 1926, in the formulation of his generalized harmonic analysis.

\[d2\] Let $t,\ q\in(0,\infty)$ and $\Phi$ be an Orlicz function with positive lower type $p_{\Phi}^-$ and positive upper type $p_{\Phi}^+$. The *Orlicz-slice space* $(E_\Phi^q)_t(\rn)$ is defined to be the set of all measurable functions $f$ such that $$\|f\|_{(E_\Phi^q)_t(\rn)}
:=\lf\{\int_{\rn}\lf[\frac{\|f\chi_{B(x,t)}\|_{L^\Phi(\rn)}}
{\|\chi_{B(x,t)}\|_{L^\Phi(\rn)}}\r]^q\,dx\r\}^{\frac{1}{q}}<\infty.$$

\[d1.3\] Let $t,\ q\in(0,\infty)$ and $\Phi$ be an Orlicz function with positive lower type $p_{\Phi}^-$ and positive upper type $p_{\Phi}^+$. The 