write $f\sim g$.

Orlicz-slice spaces\[s2\]
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In this section, we introduce the slice spaces related to Orlicz functions and present some of their basic properties such as the boundedness of maximal operators, which are used in the later sections. We begin with the notions of both Orlicz functions and Orlicz spaces (see, for example, [@mmz]).

\[d1.1\] A function $\Phi:\ [0,\infty)\ \rightarrow\ [0,\infty)$ is called an *Orlicz function* if it is non-decreasing and satisfies $\Phi(0)= 0$, $\Phi(t)>0$ whenever $t\in(0,\infty)$ and $\lim_{t\rightarrow\infty}\Phi(t)=\infty$. An Orlicz function $\Phi$ is said to be of *lower* (resp., *upper*) *type* $p$ with $p\in(-\infty,\infty)$ if there exists a positive constant $C_{(p)}$, depending on $p$, such that, for any $t\in[0,\infty)$ and $s\in(0,1)$ \[resp., $s\in [1,\infty)$\], $$\Phi(st)\le C_{(p)}s^p \Phi(t).$$ A function $\Phi:\ [0,\infty)\ \rightarrow\ [0,\infty)$ is said to be of *positive lower* (resp., *upper*) *type* if it is of lower (resp., upper) type $p$ for some $p\in(0,\infty)$.

\[d1.2\] Let $\Phi$ be an Orlicz function with positive lower type $p_{\Phi}^-$ and positive upper type $p_{\Phi}^+$. The *Orlicz space $L^\Phi(\rn)$* is defined to be the set of all measurable functions $f$ such that $$\|f\|_{L^\Phi(\rn)}:=\inf\lf\{\lambda\in(0,\infty):\ \int_{\rn}\Phi\lf(\frac{|f(x)|}{\lambda}\r)\,dx\le1\r\}<\infty.$$

We now give some basic properties of Orlicz functions.

\[lem1\] Let $\Phi$ be an Orlicz function with 