L^r(\rn)\cup L^q(\rn)$, then $f\in (E_r^q)_t(\rn)$ and $\|f\|_{(E_r^q)_t(\rn)}\leq\min\{\|f\|_{L^r(\rn)}, \|f\|_{L^q(\rn)}\}$;

-   If $q\in(0, r]$, then $(E_r^q)_t(\rn)\subset L^q(\rn)$; precisely, for any $f\in(E_r^q)_t(\rn)$, then $f\in L^q(\rn)$ and $\|f\|_{L^q(\rn)}\leq\|f\|_{(E_r^q)_t(\rn)}$.

-   $(E_q^q)_t(\rn)$ and $L^q(\rn)$ coincide with the same quasi-norms.

Observe that (iii) is an immediate corollary of (i) and (ii). Thus, to complete the proof of this proposition, we only need to show (i) and (ii).

We first show (i). In this case, for any $f\in L^q(\rn)$, using the Hölder inequality and the Fubini theorem, we obtain $$\begin{aligned}
\|f\|_{(E_r^q)_t(\rn)}&\le \lf[\int_{\rn}\lf\{\frac1{|B(x,t)|}\lf[\int_{B(x,t)}|f(y)|^q\,dy\r]^\frac rq|B(x,t)|^\frac1{(q/r)'}\r\}^\frac qr\,dx\r]^\frac1q\\
&=\lf\{\int_{\rn}\frac1{|B(x,t)|}\int_{B(x,t)}|f(y)|^q\,dy\,dx\r\}^\frac1q=\|f\|_{L^q(\rn)}.\end{aligned}$$ Also, for any $f\in L^r(\rn)$, applying the Minkowski inequality, we conclude that $$\|f\|_{(E_r^q)_t(\rn)}\le\lf[\int_{\rn}\lf\{\int_{\rn}\lf[t^{-n}\chi_{B(x,t)}(y)|f(y)|^r\r]^\frac qrdx\r\}^\frac rqdy\r]^\frac1r=\|f\|_{L^r(\rn)}.$$ Thus, $L^r(\rn)\bigcup L^q(\rn)\subset(E_r^q)_t(\rn)$, that is, (i) holds true.

Now, we prove (ii). In this case, by the Hölder inequality, we have $$\begin{aligned}
\|f\|_{L^q(\rn)}&=\lf\{\int_{\rn}\frac1{|B(x,t)|}\int_{B(x,t)}|f(y)|^q\,dy\,dx\r\}^\frac1q\\
&\leq\lf\{\int_{\rn}\frac1{|B(x,t)|}\lf[\int_{B(x,t)}|f(y)|^r\,dy\r]^{\frac{q}{r}}
\lf[\int_{B(x,t)}dy\r]^{\frac{1}{(r/q)'}}\,dx\r\}^\frac1q
=\|f\|_{(E_r^q)_t(\rn)}.\end{aligned}$$ From this, we deduce $(E_r^q)_t(\rn)\subset L^q(\rn)$, which completes the proof of (ii) and hence of Proposition \[ggg\].

Observing that, for any $x\in\rn$ and $t\in(0,\infty)$, $$\label{qiu}
\|\chi_{B(x,t)}\|_{L^\Phi(\rn)}=\lf[\Phi^{-1}\lf(\frac{1}{|B(x,t)|}\r)\r]^{-1}
=\lf[\Phi^{-1}\lf(\frac{1}{\varepsilon_nt^n}\r)\r]^{-1}=:\widetilde{C}_{(\Phi,t)}$$ is independent of $x,$ where $\varepsilon_n$ denotes the volume of the unit ball in $\rn$ and $\Phi^{-1}$ the inverse function of $\Phi$, we have the next proposition, which shows that, for any $t\in(0,\infty)$, the Orlicz-slice space $(E_\Phi^q)_t(\rn)$ is equivalent to the Orlicz-amalgam space $\ell^q(L^\Phi_t)(\rn)$.

\[th2\] 