we have $$\begin{aligned}
&\int_{\rn} \lf\{\frac{\|[\sum _{j\in\mathbb{Z}}[\mathcal{M}(f_j)]^r]^{\frac{1}{r}}\chi_{B(x,t)}\|_{L^\Phi(\rn)}}
{\|\chi_{B(x,t)}\|_{L^\Phi(\rn)}}\r\}^q\,dx\\
&\quad\lesssim \int_{\rn}\lf\{\frac{\|\{\sum_{j\in\mathbb{Z}}|f_j|^r\}^{\frac{1}{r}}
\chi_{B(x,2t)}\|_{L^\Phi(\rn)}}{\|\chi_{B(x,t)}\|_{L^\Phi(\rn)}}\r\}^q\, dx
+\int_{\rn}\lf\{\sum_{j\in\mathbb{Z}}\lf[\mathcal{M}_u\lf(\dashint_{B(\cdot,t)}|f_j(z)|dz\r)(x)\r]^r\r\}^{\frac{q}{r}}dx
\\&\quad=:\mathrm{I}+\mathrm{II}.\end{aligned}$$ Since both $\overline{B}(\vec{0}_n,2t)$ and $\overline{B}(\vec{0}_n,t)$ are compact subsets of $\rn$ with nonempty interiors, it follows that there exist $N\in\mathbb{N}$ and $\{x_1,\ \ldots,\ x_N\}\subset\rn$, independent of $t$, such that $N\lesssim1$ and $B(\vec{0}_n,2t)\subseteq\bigcup_{m=1}^NB(x_m,t)$. Thus, by this, (\[qiu\]) and the translation invariance of the Lebesgue measure, we conclude that $$\begin{aligned}
\mathrm{I}&\sim\lf[\widetilde{C}_{(\Phi,t)}\r]^q\int_{\rn}\lf\|\lf\{\sum_{j\in\mathbb{Z}}\lf|f_j\r|^r
\r\}^{\frac{1}{r}}\chi_{B(x,2t)}\r\|_{L^\Phi(\rn)}^q\, dx\\
&\lesssim\lf[\widetilde{C}_{(\Phi,t)}\r]^q\int_{\rn}\lf\|
\lf\{\sum_{j\in\mathbb{Z}}\lf|f_j\r|^r\r\}^{\frac{1}{r}}\sum_{m=1}^N\chi_{B(x+x_m,t)}\r\|_{L^\Phi(\rn)}^q\, dx\\
&\lesssim\lf[\widetilde{C}_{(\Phi,t)}\r]^q\sum_{m=1}^N\int_{\rn}
\lf\|\lf\{\sum_{j\in\mathbb{Z}}\lf|f_j\r|^r\r\}^{\frac{1}{r}}\chi_{B(x+x_m,t)}\r\|_{L^\Phi(\rn)}^q\, dx\\
&\lesssim\lf[\widetilde{C}_{(\Phi,t)}\r]^q\int_{\rn}\lf\|\lf\{\sum_{j\in\mathbb{Z}}
\lf|f_j\r|^r\r\}^{\frac{1}{r}}\chi_{B(x,t)}\r\|_{L^\Phi(\rn)}^q\, dx\sim
\lf\|\lf\{\sum _{j\in\mathbb{Z}}\lf|f_j\r|^r\r\}^{\frac{1}{r}}\r\|_{(E_\Phi^q)_t(\rn)}^q,\end{aligned}$$ where $\widetilde{C}_{(\Phi,t)}$ is as in .

As for $\mathrm{II}$, by the Fefferman-Stein vector-valued inequality in $L^q(\rn)$ (see [@fs]), we have $$\mathrm{II}
\lesssim\int_{\rn}\lf\{\sum_{j\in\mathbb{Z}}\lf[\dashint_{B(x,t)}|f_j(z)|\,dz\r]^r\r\}^{\frac{q}{r}}dx.$$ Let $r':=\frac{r}{r-1}.$ Then there exists $\{b_j\}_{j\in\mathbb{Z}}\in \ell^{r'}$, with $\|\{b_j\}_{j\in\mathbb{Z}}\|_{\ell^{r'}}=1,$ such that $$\int_{\rn}\lf\{\sum_{j\in\mathbb{Z}}\lf[\dashint_{B(x,t)}|f_j(z)|\,dz\r]^r\r\}^{\frac{q}{r}}dx
=\int_{\rn}\lf[\sum_{j\in\mathbb{Z}}b_j\dashint_{B(x,t)}|f_j(z)|\,dz\r]^{q}\,dx.$$ Using Lemma \[hold\] and the Hölder inequality, we further conclude that $$\begin{aligned}
&\int_{\rn}\lf[\sum_{j\in\mathbb{Z}}b_j\dashint_{B(x,t)}|f_j(z)|\,dz\r]^{q}\,dx\\
&\quad\lesssim\int_{\rn}\lf\{\dashint_{B(x,t)}\lf[\sum_{j\in\mathbb{Z}}|f_j(z)|^r\r]^{\frac{1}{r}}
\lf(\sum_{j\in\mathbb{Z}}b_j^{r'}\r)^{\frac{1}{r'}}\,dz\r\}^{q}\,dx\\
&\quad\lesssim\int_{\rn}\lf\{\lf\|\lf[\sum_{j\in\mathbb{Z}}\lf|f_j\r|^r\r]^{\frac{1}{r}}
\chi_{B(x,t)}\r\|_{L^\Phi(\rn)}\frac{\|\chi_{B(x,t)}\|_{L^\Psi(\rn)}}{|B(x,t)|}\r\}^q\, dx.\end{aligned}$$ Applying Lemma \[un\], we obtain $$\begin{aligned}
\frac{\|\chi_{B(x,t)}\|_{L^\Psi(\rn)}}{|B(x,t)|}
&=\frac{1}{|B(x,t)|\Psi^{-1}(\frac{1}{|B(x,t)|})}
=\frac{\Phi^{-1}(\frac{1}{|B(x,t)|})}{|B(x,t)|\Phi^{-1}(\frac{1}{|B(x,t)|})\Psi^{-1}(\frac{1}{|B(x,t)|})}\\
&<\Phi^{-1}\lf(\frac{1}{|B(x,t)|}\r)=\frac{1}{\|\chi_{B(x,t)}\|_{L^\Phi(\rn)}}.\end{aligned}$$ Thus, $$\mathrm{II}\lesssim\lf\|\lf\{\sum _{j\in\mathbb{Z}}|f_j|^r\r\}^{\frac{1}{r}}\r\|_{(E_\Phi^q)_t(\rn)}^q,$$ which completes the proof of Theorem \[main\].

Let $t\in(0,\infty),$ $q\in(1,\infty)$ and $\Phi(\tau):=\tau^q$ for any $\tau\in[0,\infty).$ Then, by Remark \[re2.10\](ii) and Proposition \[ggg\](iii), we know that $(E_\Phi^q)_t(\rn)=L^q(\rn)$ and, in this case, Theorem \[main\] is just the well-known Fefferman-Stein vector-valued maximal inequality [@fs Theorem 1(1)].

As an immediate consequence of Theorem \[main\], we have the following boundedness of Hardy-Littlewood maximal operators on 