and ${\overrightarrow{q}}={\left(}q_i{\right)}_{i\in\mathcal{I}_1\cup\mathcal{I}_2}$ be such that $q_i>1$ for all $i\in\mathcal{I}_1\cup\mathcal{I}_2$. The idea is to apply Lemma \[Lem4\] by induction. For any $\gamma>p_*-1$, Lemma \[Lem4\] provides an estimate of the $L^\gamma$–norm of $u$ with respect to the set of $L^{\gamma_{i_1}}$–norms of $u$, where $\gamma_{i_1}:=\frac{n-p}{n}\gamma+p_{i_1}-p$ for all $i_1\in\mathcal{I}_1\cup\mathcal{I}_2$. If $\gamma_{i_1}>p_*-1$, then another application of Lemma \[Lem4\] gives estimates of the $L^{\gamma_{i_1}}$–norms of $u$ with respect to the set of $L^{\gamma_{i_1i_2}}$–norms of $u$, where $\gamma_{i_1i_2}:=\frac{n-p}{n}\gamma_{i_1}+p_{i_2}-p$, etc... By induction, we define $$\label{Eq11}
\gamma_{i_1,\dotsc,i_{j+1}}:=\frac{n-p}{n}\gamma_{i_1,\dotsc,i_j}+p_{i_{j+1}}-p$$ for all $j\in{\mathbb{N}}$ and $i_1,\dotsc,i_{j+1}\in\mathcal{I}_1\cup\mathcal{I}_2$, with the convention that $\gamma_{i_1,\dotsc,i_j}:=\gamma$ if $j=0$. In particular, we obtain the formula $$\label{Eq12}
\gamma_{i_1,\dotsc,i_k}={\left(}\frac{n-p}{n}{\right)}^k\gamma+\sum_{j=1}^k{\left(}\frac{n-p}{n}{\right)}^{k-j}{\left(}p_{i_j}-p{\right)}$$ for all $k\in{\mathbb{N}}$. The stopping condition in our induction argument is $\gamma_{i_1,\dotsc,i_k}<\frac{n}{p}{\left(}p_\varepsilon-p{\right)}$, where $$\label{Eq13}
p_\varepsilon:={\left(}1+\varepsilon{\right)}p_0\,,\quad p_0:=\max{\left(}p_*,\left\{p_i\,:\,i\in\mathcal{I}_1\cup\mathcal{I}_2\right\}{\right)},$$ and $\varepsilon$ is a fixed real number in ${\left(}0,1{\right)}$. Note that $\frac{n}{p}{\left(}p_\varepsilon-p{\right)}>\frac{n}{p}{\left(}p_*-p{\right)}=p_*-1$ so that we can apply Lemma \[Lem4\] as long as our stopping condition is not satisfied. For any $k\ge1$, we let $\Phi_{k,\gamma,\varepsilon}$ be the set of all sequences of indices for which our induction argument stops after exactly $k$ iterations, namely $$\begin{gathered}
\label{Eq14}
\Phi_{k,\gamma,\varepsilon}:=\Big\{{\left(}i_1,\dotsc,i_k{\right)}\in{\left(}\mathcal{I}_1\cup\mathcal{I}_2{\right)}^k\,:\quad\gamma_{i_1,\dotsc,i_j}\ge\frac{n}{p}{\left(}p_\varepsilon-p{\right)}\text{ for all }j=0,\dotsc,k-1\\
\text{and}\quad\gamma_{i_1,\dotsc,i_k}<\frac{n}{p}{\left(}p_\varepsilon-p{\right)}\Big\}.\end{gathered}$$ The following result provides a control on the number of iterations in our induction argument.

\[Lem6\] Let $\mathcal{I}_1$ and $\mathcal{I}_2$ be two disjoint subsets of $\left\{1,\dotsc,n\right\}$, 