where $\delta_{i_j}:=1$ if $i_j\in\mathcal{I}_1$, $\delta_{i_j}:=2$ if $i_j\in\mathcal{I}_2$, $\gamma_{i_1,\dotsc,i_k}$ is as in , $p_\varepsilon$ is as in , $\Omega_{{\overrightarrow{q}}}{\left(}\mathcal{I}_1,R_1,\mathcal{I}_2,R_2,\lambda_k{\right)}$ is as in , and $$\label{Lem7Eq2}
\lambda_{k,\gamma,\varepsilon}:=\frac{1}{4}\big(1+2^{k-k_{\gamma,\varepsilon}^+-1}\big)\quad\text{and}\quad\Phi_{\gamma,\varepsilon}:=\bigcup_{k=k_{\gamma,\varepsilon}^-}^{k_{\gamma,\varepsilon}^+}\Phi_{k,\gamma,\varepsilon}$$ with $k_{\gamma,\varepsilon}^-$ and $k_{\gamma,\varepsilon}^+$ as in Lemma \[Lem6\], and $\Phi_{k,\gamma,\varepsilon}$ as in .

Applying Lemma \[Lem4\] by induction with the stopping condition $\gamma_{i_1,\dotsc,i_k}<\frac{n}{p}{\left(}p_\varepsilon-p{\right)}$ gives $$\begin{gathered}
\label{Lem7Eq3}
\left\|u\right\|_{L^{\gamma}{\left(}\Omega_{{\overrightarrow{q}}}{\left(}\mathcal{I}_1,R_1,\mathcal{I}_2,R_2,\lambda_{0,\gamma,\varepsilon}{\right)}{\right)}}\le\max_{{\left(}i_1,\dotsc,i_k{\right)}\in\Phi_{\gamma,\varepsilon}}\Big(\mathcal{A}_{k,\gamma}\times\mathcal{B}_{i_1,\dotsc,i_{k-1},\gamma}\times\mathcal{C}_{i_1,\dotsc,i_k,\gamma}\times\mathcal{D}_{i_1,\dotsc,i_k,\gamma,\varepsilon}\\
\times{\left(}\prod_{j=1}^kR_{\delta_{i_j}}^{-\frac{1}{\gamma}{\left(}\frac{n}{n-p}{\right)}^j\frac{p_{i_j}}{q_{i_j}}}{\right)}\left\|u\right\|_{L^{\gamma_{i_1,\dotsc,i_k}}{\left(}\Omega_{{\overrightarrow{q}}}{\left(}\mathcal{I}_1,R_1,\mathcal{I}_2,R_2,\lambda_{k,\gamma,\varepsilon}{\right)}{\right)}}^{\frac{\gamma_{i_1,\dotsc,i_k}}{\gamma}{\left(}\frac{n}{n-p}{\right)}^k}\Big),\end{gathered}$$ where $\lambda_{k,\gamma,\varepsilon}$ and $\Phi_{\gamma,\varepsilon}$ are as in , and $$\begin{aligned}
&\mathcal{A}_{k,\gamma}:=\big(c_0\cdot\max\big(q_i^{\frac{np_i}{n-p}}\big)\big)^{\frac{1}{\gamma}\overset{k-1}{\underset{j=0}{\sum}}{\left(}\frac{n}{n-p}{\right)}^j},\quad\mathcal{B}_{i_1,\dotsc,i_{k-1},\gamma}:=\prod_{j=1}^k\gamma_{i_1,\dotsc,i_{j-1}}^{\frac{p}{\gamma}{\left(}\frac{n}{n-p}{\right)}^j},\allowdisplaybreaks\\
&\mathcal{C}_{i_1,\dotsc,i_{k-1},\gamma}:=\prod_{j=1}^k\min{\left(}1,\gamma_{i_1,\dotsc,i_{j-1}}-p_*+1{\right)}^{-\frac{p_{i_j}}{\gamma}{\left(}\frac{n}{n-p}{\right)}^j},\allowdisplaybreaks\\
&\mathcal{D}_{i_1,\dotsc,i_k,\gamma,\varepsilon}:=\prod_{j=1}^k{\left(}\lambda_{j,\gamma,\varepsilon}-\lambda_{j-1,\gamma,\varepsilon}{\right)}^{-\frac{p_{i_j}}{\gamma}{\left(}\frac{n}{n-p}{\right)}^j}.\end{aligned}$$ Now, we fix ${\left(}i_1,\dotsc,i_k{\right)}\in\Phi_{\gamma,\varepsilon}$ and estimate each of the terms in the right-hand side of .

By using the fact that $k\le k_{\gamma,\varepsilon}^+$ and applying , we obtain $$\label{Lem7Eq4}
\sum_{j=0}^{k-1}{\left(}\frac{n}{n-p}{\right)}^j=\frac{n-p}{p}{\left[}{\left(}\frac{n}{n-p}{\right)}^k-1{\right]}<\frac{n}{p}{\left(}\frac{n}{n-p}{\right)}^{k_{\gamma,\varepsilon}^+-1}<\frac{\gamma}{\varepsilon p_0}\,.$$ Since $c_0>1$, $q_i>1$, and $p_i\le p_0$ for all $i\in\mathcal{I}_1\cup\mathcal{I}_2$, it follows from that $$\label{Lem7Eq5}
\mathcal{A}_{k,\gamma}<c_0^{\frac{1}{\varepsilon p_0}}\max{\left(}q_i{\right)}^{\frac{1}{\varepsilon}\cdot\frac{n}{n-p}}.$$

For any $j=1,\dotsc,k$, since $p_i\le p_0$ for all $i\in\mathcal{I}_1\cup\mathcal{I}_2$, by , , and , we obtain $$\label{Lem7Eq6}
\gamma_{i_1,\dotsc,i_{j-1}}\le{\left(}\frac{n-p}{n}{\right)}^{j-1}\gamma+\frac{n}{p}{\left(}p_0-p{\right)}\le C\max\bigg(1,{\left(}\frac{n}{n-p}{\right)}^{k_{\gamma,\varepsilon}^--j}\bigg)$$ for some constant $C=C{\left(}n,\operatorname{\overrightarrow{p}}{\right)}>1$. It follows from that $$\label{Lem7Eq7}
\mathcal{B}_{i_1,\dotsc,i_{k-1},\gamma}\le C^{\frac{p}{\gamma}\overset{k}{\underset{j=1}{\sum}}{\left(}\frac{n}{n-p}{\right)}^j}{\left(}\frac{n}{n-p}{\right)}^{\frac{p}{\gamma}\overset{k_{\gamma,\varepsilon}^-}{\underset{j=1}{\sum}}{\left(}\frac{n}{n-p}{\right)}^j{\left(}k_{\gamma,\varepsilon}^--j{\right)}}.$$ A simple calculation gives $$\overset{k_{\gamma,\varepsilon}^-}{\underset{j=1}{\sum}}{\left(}\frac{n}{n-p}{\right)}^j{\left(}k_{\gamma,\varepsilon}^--j{\right)}=\frac{n^2}{p^2}{\left[}{\left(}\frac{n}{n-p}{\right)}^{k_{\gamma,\varepsilon}^--1}-\frac{p}{n}{\left(}k_{\gamma,\varepsilon}^--1{\right)}-1{\right]}<\frac{n^2}{p^2}{\left(}\frac{n}{n-p}{\right)}^{k_{\gamma,\varepsilon}^--1},$$ and hence by definition of $k_{\gamma,\varepsilon}^-$, we obtain $$\label{Lem7Eq8}
\overset{k_{\gamma,\varepsilon}^-}{\underset{j=1}{\sum}}{\left(}\frac{n}{n-p}{\right)}^j{\left(}k_{\gamma,\varepsilon}^--j{\right)}\le\frac{n}{p}\cdot\frac{\gamma}{p_\varepsilon-p_-}<\frac{n}{p}\cdot\frac{\gamma}{p_*-p_-}\,.$$ It follows from , , and that $$\label{Lem7Eq9}
\mathcal{B}_{i_1,\dotsc,i_{k-1},\gamma}\le C^{\frac{p^*}{\varepsilon p_0}}{\left(}\frac{n}{n-p}{\right)}^{\frac{n}{p_*-p_-}}.$$

Since $p_*-1=\frac{n}{p}{\left(}p_*-p{\right)}$ and $\gamma_{i_1,\dotsc,i_{j-1}}>\frac{n}{p}{\left(}p_\varepsilon-p{\right)}$ for all $j=1,\dotsc,k$, we obtain $$\label{Lem7Eq10}
\mathcal{C}_{i_1,\dotsc,i_{k-1},\gamma}\le\min\Big(1,\frac{n}{p}{\left(}p_\varepsilon-p_*{\right)}\Big)^{-\frac{1}{\gamma}\overset{k}{\underset{j=1}{\sum}}p_{i_j}{\left(}\frac{n}{n-p}{\right)}^j}.$$ Since $p_{i_j}\le p_0$ for all $j=1,\dotsc,k$, it follows from and that $$\label{Lem7Eq11}
\mathcal{C}_{i_1,\dotsc,i_{k-1},\gamma}\le\min\Big(1,\frac{n}{p}{\left(}p_\varepsilon-p_*{\right)}\Big)^{-\frac{1}{\varepsilon}\cdot\frac{n}{n-p}}.$$

By and since $k\le k_{\gamma,\varepsilon}^+$ and $p_{i_j}\le 