functions and $x\in\rn$, it holds true that $$\begin{aligned}
\lf\|\lf\{\sum _{j\in\mathbb{Z}}\lf[\mathcal{M}(f_j)\r]^r\r\}^{\frac{1}{r}}\chi_{B(x,t)}\r\|_{L^\Phi(\rn)}
&\le C\lf\|\lf\{\sum_{j\in\mathbb{Z}}\lf|f_j\r|^r\r\}^{\frac{1}{r}}\chi_{B(x,2t)}\r\|_{L^\Phi(\rn)}
\\&\quad+C\|\chi_{B(x,t)}\|_{L^\Phi(\rn)}
\lf\{\sum_{j\in\mathbb{Z}}\lf[\mathcal{M}_u\lf(\dashint_{B(\cdot,t)}\lf|f_j(z)\r|\,dz\r)(x)\r]^r\r\}^{\frac{1}{r}},\end{aligned}$$ where the positive constant $C$ is independent of $\{f_j\}_{j\in\mathbb{Z}}$, $x\in\rn$ and $t\in(0,\infty)$.

Let $\{f_j\}_{j\in\mathbb{Z}}$ be a sequence of locally integrable functions. Fix $x\in\rn$. Then we have

$$\begin{aligned}
\lf\|\lf\{\sum _{j\in\mathbb{Z}}\lf[\mathcal{M}(f_j)\r]^r\r\}^{\frac{1}{r}}\chi_{B(x,t)}\r\|_{L^\Phi(\rn)}
&\lesssim\lf\|\lf\{\sum _{j\in\mathbb{Z}}\lf[\sup_{s\in(0, t]}\dashint_{B(\cdot,s)}\lf|f_j(z)\r|\,dz\r]^r\r\}^{\frac{1}{r}}\chi_{B(x,t)}\r\|_{L^\Phi(\rn)}\\
&\quad+\lf\|\lf\{\sum _{j\in\mathbb{Z}}\lf[\sup_{s\in(t,\infty)}\dashint_{B(\cdot,s)}\lf|f_j(z)\r|\,dz\r]^r
\r\}^{\frac{1}{r}}\chi_{B(x,t)}\r\|_{L^\Phi(\rn)}\\
&=:\mathrm{I}+\mathrm{II}.\end{aligned}$$

Since $B(y,s)\subset B(x,2t)$ whenever $s\in(0,t]$ and $y\in B(x,t)$, it follows that $$\begin{aligned}
\mathrm{I}&\sim\lf\|\chi_{B(x,t)}\lf\{\sum _{j\in\mathbb{Z}}\lf[\sup_{s\in(0, t]}\dashint_{B(\cdot,s)}\lf|f_j(z)\r|\chi_{B(x,2t)}(z)\,dz\r]^r\r\}^{\frac{1}{r}}\r\|_{L^\Phi(\rn)}\\
&\lesssim\lf\|\lf\{\sum _{j\in\mathbb{Z}}\lf[\mathcal{M}\lf(f_j\chi_{B(x,2t)}\r)\r]^r\r\}^{\frac{1}{r}}\r\|_{L^\Phi(\rn)}
\lesssim\lf\|\lf\{\sum _{j\in\mathbb{Z}}\lf|f_j\r|^r\r\}^{\frac{1}{r}}\chi_{B(x,2t)}\r\|_{L^\Phi(\rn)},\end{aligned}$$ where, in the last inequality, we used the Orlicz Fefferman-Stein vector-valued inequality (see, for example, [@kk] or [@yll Theorem 2.1.4]).

As for $\mathrm{II}$, observe that, for any $\xi,\ z\in\rn$, $\xi\in B(z,t)$ if and only if $z\in B(\xi,t)$ and, moreover, if $z\in B(y,s)$, $\xi\in B(z,t)$ and $s\in(t,\infty)$, then $\xi\in B(y,2s)$. Besides, observe that $y\in B(x,t)$ and $s\in(t,\infty)$ imply that $x\in B(y,2s).$ From these observations and the Fubini theorem, we deduce that $$\begin{aligned}
\mathrm{II}&\sim\lf\|\chi_{B(x,t)}\lf\{\sum_{j\in\mathbb{Z}}\lf[\sup_{s\in(t,\infty)}
\dashint_{B(\cdot,s)}\dashint_{B(z,t)}\lf|f_j(z)\r|
\,d\xi \,dz\r]^r\r\}^{\frac{1}{r}}\r\|_{L^\Phi(\rn)}\\
&\lesssim\lf\|\chi_{B(x,t)}\lf\{\sum_{j\in\mathbb{Z}}\lf[\sup_{s\in(t,\infty)}
\dashint_{B(\cdot,2s)}\dashint_{B(\xi,t)}\lf|f_j(z)\r|\,dz\,d\xi\r]^r\r\}^{\frac{1}{r}}\r\|_{L^\Phi(\rn)}\\
&\lesssim\lf\|\chi_{B(x,t)}\lf\{\sum_{j\in\mathbb{Z}}
\lf[\mathcal{M}_u\lf(\dashint_{B(\cdot,t)}\lf|f_j(z)\r|\,dz\r)(x)\r]^r\r\}^{\frac{1}{r}}\r\|_{L^\Phi(\rn)}\\
&\lesssim\lf\|\chi_{B(x,t)}\r\|_{L^\Phi(\rn)}\lf\{\sum_{j\in\mathbb{Z}}
\lf[\mathcal{M}_u\lf(\dashint_{B(\cdot,t)}\lf|f_j(z)\r|\,dz\r)(x)\r]^r\r\}^{\frac{1}{r}}.\end{aligned}$$

Gathering the estimates for $\mathrm{I}$ and $\mathrm{II}$, we then obtain the desired conclusion, which completes the proof of Lemma \[mm\].

\[d1.4\] A convex function $\Phi:\ [0,\infty)\  \rightarrow\  [0,\infty)$ is called a *Young function* if $\Phi$ is non-decreasing, $\Phi(0)= 0$ and $\lim_{t\rightarrow\infty}\Phi(t)=\infty$. For any Young function $\Phi$, its complementary function $\Psi:\ 