By hypothesis $V_n \geq 0$ in $\Omega$ and $\lVert V_n \rVert_{L^{p}} \leq C_1$ for some $1<p \leq \infty$.

Why $k$ large enough implies that some neighborhood contains $z$? It can happens that the neighboors as $k$ gets larger are smaller and smaller always avoiding $z$

Why is it possible to apply Calderón - Zygmund so directly in $L^p$? I mean, following corollary 9.18 of trudinger it's possible to apply the estimate only when $f_n- \mu \in L^p$ for $p>1$, but in the situation of the theorem we have $f_n-\mu \in L^{p}_{loc}$

Why is enough to show that $u_3^{+} \in L^{\infty}$ to conclude that $u_3 \in L^{\infty}$? why the proof ends when we show that $u^{+} \in L^{\infty}?$ It's not necessary to show this to $u^{-} \in L^{\infty}$ also?

that is exactly what i'm looking for! thank you!

Thank you @NoahSchweber, now I have access to the article, and I think this is just what I'm looking for :) Time to read