to complete the proof of Proposition \[th2\], we only need to show that $$\lf\|\lf\|f\chi_{B(\cdot,2\sqrt{n}t)}\r\|_{L^\Phi(\rn)}\r\|_{L^q(\rn)}
\backsim\lf\|\lf\|f\chi_{B(\cdot,t)}\r\|_{L^\Phi(\rn)}\r\|_{L^q(\rn)}.$$ Since both $\overline{B}(\vec{0}_n,4\sqrt{n}t)$ and $\overline{B}(\vec{0}_n,t)$ are compact subsets of $\rn$ with nonempty interiors, it follows that there exist $M\in\mathbb{N}$ and $\{x_1,\ \ldots,\ x_M\}\subset\rn$, independent of $t$, such that $M\lesssim1$ and $B(\vec{0}_n,4\sqrt{n}t)\subseteq\bigcup_{m=1}^MB(x_m,t).$ Thus, for any $x\in\rn$, we have $$\begin{aligned}
\lf\|f\chi_{B(x,2\sqrt{n}t)}\r\|_{L^\Phi(\rn)}&=\lf\|f\sum_{m=1}^M\chi_{B(x+x_m,t)}\r\|_{L^\Phi(\rn)}
\lesssim\sum_{m=1}^M\lf\|f\chi_{B(x+x_m,t)}\r\|_{L^\Phi(\rn)}.\end{aligned}$$ By this, the translation invariance of the Lebesgue measure and $M\lesssim1$, we further obtain $$\lf\|\lf\|f\chi_{B(\cdot,2\sqrt{n}t)}\r\|_{L^\Phi(\rn)}\r\|_{L^q(\rn)}
\lesssim\sum_{m=1}^M\lf\|\lf\|f\chi_{B(x+x_m,t)}\r\|_{L^\Phi(\rn)}\r\|_{L^q(\rn)}
\lesssim\lf\|\lf\|f\chi_{B(\cdot,t)}\r\|_{L^\Phi(\rn)}\r\|_{L^q(\rn)}.$$ The reverse inequality obviously holds true. This finishes the proof of Proposition \[th2\].

Recall that the *centered Hardy-Littlewood maximal operator* $\mathcal{M}$ is defined by setting, for any locally integrable function $f$ and $x\in\rn$, $$\mathcal{M}(f)(x):=\sup_{r\in(0,\infty)}\dashint_{B(x,r)}|f(x)|\,dy,$$ and the *uncentered Hardy-Littlewood maximal operator $\mathcal{M}_u$* is defined by setting, for any locally integrable function $f$ and $x\in\rn$, $$\mathcal{M}_u(f)(x):=\sup_{x\in B} \dashint_B |f(y)|\,dy,$$ where the supremum is taken over all balls $B$ of $\rn$ containing $x$.

Borrowing some ideas from the proof of [@ap Lemma 4.1], we have the following very useful technical lemma, which plays a vital role in the proof of Theorem \[main\] below.

\[mm\] Let $t\in(0,\infty),\ r\in(1,\infty)$ and $\Phi$ be an Orlicz function with positive lower type $p_{\Phi}^-\in(1,\infty)$ and positive upper type $p_{\Phi}^+$. Then, for any sequence $\{f_j\}_{j\in\mathbb{Z}}$ of locally integrable 