Orlicz-slice spaces (see Theorem \[dual2\] below).

The last section, Section \[s6\], is devoted to some further applications of the characterizations obtained in Section \[s3\]. We first establish a criterion on the boundedness of sublinear operators from $(HE_\Phi^q)_t(\rn)$ into a quasi-Banach space (see Theorem \[suanzi1\] and Corollary \[suanzi\] below), which are further used to obtain the boundedness of the $\delta$-type Calderón-Zygmund operators on $(HE_\Phi^q)_t(\rn)$ (see Theorems \[cz\] and \[cz2\]). Moreover, in Proposition \[au\] below, we clarify the relation between $(\mathfrak{C}_r^q)_t(\rn)$, with $t\in (0,\fz)$, $r\in (1,\fz)$ and $q\in (\frac n{n+1}, 1]$, from [@ap] and $(HE_\Phi^q)_t(\rn)$ in the case when $\Phi(\tau):=\tau^s$ for any $\tau\in [0,\fz)$ with $s\in (0,q]$.

Observe that a real-variable theory of local Hardy spaces based on the Orlicz-slice spaces can also be developed. However, to limit the length of this article, we will consider this local version in another article.

Finally, we make some convention on notation. For any $x\in\rn$ and $r\in(0,\infty)$, let $B(x,r):=\{y\in\rn:|x-y|<r\}$ and $\bar{B}(x,r)$ be its *closure* in $\rn$. For any $r\in(0,\infty)$, $f\in L^1_\loc(\rn)$ and $x\in\rn$, let $$\dashint_{B(x,\,r)}f(y)\,dy:=\frac1{|B(x,r)|}\int_{B(x,r)}f(y)\,dy,$$ here and hereafter, $L^1_\loc(\rn)$ denotes the space of all locally integrable functions. For any set $E$, we use $\chi_{E}$ to denote its *characteristic function* and $\# E$ its *cardinality*. We 