I'm struggling with an application of Sobolev inequalities in Evans' book.
He presents his argument like this:
For $4<p<5$ we have $2(p-1)=2(p-4)+6=2(p-4)+ 2^*$ and therefore $$\left( \int_{B(x,t)}|u^*|^{2(p-1)} dy \right)^\frac{1}{2}\leq C\lVert u\rVert_{L^\infty}^{p-4}$$ where $u^*(y):=u(y,t-|y-x|)$ (which shouldn't be that relevant I think).
This is for $n=3$, so for any $p\geq 1$, $p^*$ is given by $\frac{1}{p^*}=\frac{1}{p}-\frac{1}{3}$.
Here's a compact version of the Theorem he's referring to:
$U\subseteq \mathbb{R}^n$ bounded & open with $C^1$-boundary, $u\in W^{k,p}(U)$. If $k<\frac{n}{p}$, then $u\in L^q(U)$ where $\frac{1}{q}=\frac{1}{p}-\frac{k}{n}$ and we have $\lVert u\rVert_{L^q(U)}\leq C\lVert u\rVert_{W^{k,p}(U)}$.
I understand that for $k=1$, $q$ is $p^*$ and I also tried $$\left( \int_{B(x,t)}|u^*|^{2(p-1)} dy \right)^\frac{1}{2}=||u^*||_{L^{2(p-1)}(B(x,t))}^{p-1}$$ or applying Hölder's inequality to $\lVert {u^*}^{2(p-4)}{u^*}^{2^*}\rVert_{L^1}$, but I don't make any progress towards that $L^\infty$-estimate... He probably uses the Sobolev inequality together with some estimate which was found earlier in the proof, but I don't see anything relevant, so I want to concentrate on the Sobolev inequality first. So my question is:
How to apply an appropiate Sobolev inequality here if you know that $2(p-1)=2(p-4)+2^*$?