Let $t,\ q\in(0,\infty)$ and $\Phi$ be an Orlicz function with positive lower type $p_{\Phi}^-$ and positive upper type $p_{\Phi}^+$. Then $(E_\Phi^q)_t(\rn)$ and $\ell^q(L^\Phi_t)(\rn)$ coincide and, for any $f\in(E_\Phi^q)_t(\rn)$, $$\lf[t^{n}\sum_{k\in \mathbb{Z}^n}\lf\|f\chi_{Q_{  kt}}\r\|^q_{L^\Phi(\rn)}\r]^{\frac{1}{q}}
\sim\lf\{\int_{\rn}\lf[\lf\|f\chi_{B(x,t)}\r\|_{L^\Phi(\rn)}\r]^q\,dx\r\}^{\frac{1}{q}},$$ where the equivalent positive constants are independent of $f$ and $t$.

We first show that $$\label{dj}
\lf[t^{n}\sum_{k\in \mathbb{Z}^n}\lf\|f\chi_{Q_{tk}}\r\|^q_{L^\Phi(\rn)}\r]^{\frac{1}{q}}
\sim\lf\{\int_{\rn}\lf[\lf\|f\chi_{B(x,2\sqrt{n}t)}\r\|_{L^\Phi(\rn)}\r]^q\,dx\r\}^{\frac{1}{q}}.$$ Indeed, it is easy to see that $$\begin{aligned}
\lf[t^{n}\sum_{k\in \mathbb{Z}^n}\lf\|f\chi_{Q_{tk}}\r\|^q_{L^\Phi(\rn)}\r]^{\frac{1}{q}}
=\lf\|\sum_{k\in \mathbb{Z}^n}\lf\|f\chi_{Q_{tk}}\r\|_{L^\Phi(\rn)}\chi_{Q_{tk}}\r\|_{L^q(\rn)}.\end{aligned}$$ For any $x\in Q_{tk}$, from $Q_{tk}\subset B(x,2\sqrt{n}t)$, it follows that $$\lf\|f\chi_{Q_{tk}}\r\|_{L^\Phi(\rn)}\chi_{Q_{tk}}(x)\le\lf\|f\chi_{B(x,2\sqrt{n}t)}\r\|_{L^\Phi(\rn)}.$$ Thus, combining the above two formulas, we conclude that

$$\label{yyy}
\lf\|\sum_{k\in \mathbb{Z}^n}\lf\|f\chi_{Q_{tk}}\r\|_{L^\Phi(\rn)}\chi_{Q_{tk}}\r\|_{L^q(\rn)}
\le\lf\|\lf\|f\chi_{B(\cdot,2\sqrt{n}t)}\r\|_{L^\Phi(\rn)}\r\|_{L^q(\rn)}.$$

To prove the opposite inequality, for any given $x\in \rn$, we let $$M_x:=\lf\{k\in\mathbb{Z}^n:\ Q_{tk} \cap B(x,2\sqrt{n}t)\neq\emptyset\r\}.$$ Then the cardinality $\#M_x\lesssim1$ and, if $k\in M_x$, then $x\in B(tk,4\sqrt{n}t)$, which further implies that $$\begin{aligned}
\lf\|f\chi_{B(x,2\sqrt{n}t)}\r\|_{L^\Phi(\rn)}&=\lf\|\sum_{k\in \mathbb{Z}^n}f
\chi_{B(x,2\sqrt{n}t)}\chi_{Q_{tk}}\r\|_{L^\Phi(\rn)}
\lesssim\sum_{k\in M_x}\lf\|f\chi_{B(x,2\sqrt{n}t)}\chi_{Q_{tk}}\r\|_{L^\Phi(\rn)}\\
&\lesssim\sum_{k\in \mathbb{Z}^n}\lf\|f\chi_{Q_{tk}}\r\|_{L^\Phi(\rn)}\chi_{B(tk,4\sqrt{n}t)}(x).\end{aligned}$$ Thus, we have $$\label{yy}
\lf\|\lf\|f\chi_{B(\cdot,2\sqrt{n}t)}\r\|_{L^\Phi(\rn)}\r\|_{L^q(\rn)}\lesssim\lf\|\sum_{k\in \mathbb{Z}^n}\lf\|f\chi_{Q_{tk}}\r\|_{L^\Phi(\rn)}\chi_{B(tk,4\sqrt{n}t)}\r\|_{L^q(\rn)}.$$ It is easy to see that there exist $N\in\mathbb{N}$ and $\{k_1,\ \ldots,\ k_N\}\subset\mathbb{Z}^n$, independent of $t$, such that $N\lesssim1$ and $B(\vec{0}_n,4\sqrt{n}t)$ $\subseteq\bigcup_{m=1}^N Q_{tk_m}$ and hence $$\sum_{k\in \mathbb{Z}^n}\lf\|f\chi_{Q_{tk}}\r\|_{L^\Phi(\rn)}\chi_{B(tk,4\sqrt{n}t)}
\le\sum_{m=1}^N\sum_{k\in \mathbb{Z}^n}\lf\|f\chi_{Q_{tk}}\r\|_{L^\Phi(\rn)}\chi_{Q_{t(k_m+k)}}.$$ By this, the translation invariance of the Lebesgue measure and $N\lesssim1$, we further obtain $$\begin{aligned}
\lf\|\sum_{k\in \mathbb{Z}^n}\lf\|f\chi_{Q_{tk}}\r\|_{L^\Phi(\rn)}\chi_{B(tk,4\sqrt{n}t)}\r\|_{L^q(\rn)}
&\leqslant\lf\|\sum_{m=1}^N\sum_{k\in \mathbb{Z}^n}\lf\|f\chi_{Q_{tk}}\r\|_{L^\Phi(\rn)}\chi_{Q_{t(k_m+k)}}\r\|_{L^q(\rn)}\\
&\lesssim\lf[t^{n}\sum_{k\in \mathbb{Z}^n}\lf\|f\chi_{Q_{tk}}\r\|^q_{L^\Phi(\rn)}\r]^{\frac{1}{q}},\end{aligned}$$ which, together with (\[yy\]), implies that the opposite inequality of (\[yyy\]) holds true. Thus, (\[dj\]) holds true.

Now, 