 \, \|v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)} - w_{(\xi,\varepsilon)}\|_{L^{\frac{0n}{n-0}}(\mathbb{R}^n)} \\ &\leq C \, \lambda^{\frac{00n}{n-0}} \, \mu^{\frac{0n}{n-0}} + C \, \lambda^0 \, \mu \, \Big ( \frac{\lambda}{\rho} \Big )^{\frac{n-0}{0}}\end{aligned}$$ by Corollary \[estimate.for.v.0\]. Putting these facts together, the assertion follows.\  \[term.0\] We have $$\begin{aligned} &\bigg | \int_{\mathbb{R}^n} (|v_{(\xi,\varepsilon)}|^{\frac{0}{n-0}} - u_{(\xi,\varepsilon)}^{\frac{0}{n-0}}) \, u_{(\xi,\varepsilon)} \, v_{(\xi,\varepsilon)} - \frac{0}{n} \int_{\mathbb{R}^n} (|v_{(\xi,\varepsilon)}|^{\frac{0n}{n-0}} - u_{(\xi,\varepsilon)}^{\frac{0n}{n-0}}) \bigg | \\ &\leq C \, \lambda^{\frac{00n}{n-0}} \, \mu^{\frac{0n}{n-0}} + C \, \Big ( \frac{\lambda}{\rho} \Big )^n\end{aligned}$$ whenever $(\xi,\varepsilon) \in \lambda \, \Omega$.  **Proof.** 