Sobolev's inequality for $nI Have two questions:
$1)$Let $n<p<\infty$ and $|U|<\infty$. Then there exist $C=C(n,p)$ such that $$ess\sup_U |u|\le C|U|^{(p-n)/pn}\left( \int_U |Du|^p\right)^{1/p}$$ for any $u\in W_0^{1,p}(U)$.
My attempt: I want use the Morrey's inequality $$\|u\|_{C^{0,\gamma}(\Bbb R^n)}\le C\|u\|_{W^{1,p}(\Bbb R^n)},$$ where $n<p<\infty$ and the costant $C$ depends only $n$ and $p$. I want use also the Poincare's inequality $$\int_U |u|^pdx\le C^p diam(U)^p\int_U |Du|^pdx,$$ for every $u\in W_0^{1,p}(U)$ and the costant $C$ depend only by $n$. The problem is that i use the definition of $W^{1,p}_0(U)$ then i have the diameter and not the term $|U|$.
$2)$ Let $p=n>1$. Then for any $q\in (0,\infty)$ there exist $C=C(n,q)$ such that $$\left( \int_{B(x,r)} |u|^qdy\right) ^{1/q}\le Cr^{p/q} \left( \int_{B(x,r)} |Du|^pdy\right )^{1/p},$$ for $u\in W^{1,p}_0(B(x,r)).$
How i can prove these two theorems?
 A: We do 1) first. By a scaling argument, it's enough to prove the estimate for all $U$ with $|U|=1$. For this, we use the representation formula
$$
u(x)=c\int_{\mathbb{R}^n} \dfrac{\nabla u(y)\cdot (x-y)}{|x-y|^n}\, dy, \qquad x\in \mathbb{R}^n,
$$
valid for all smooth compactly supported functions.
In particular, if $u\in C_c^\infty(U)$, then by Hölder's inequality
$$
|u(x)|\leq \| \nabla u\|_{L^p(U)} \left\| |x-\cdot|^{1-n}\right\|_{L^q(U)}, \qquad \frac{1}{p}+\frac{1}{q}=1.
$$
It remains to show that the $L^q$ norm is bounded uniformly in $x$; notice that $p>n$ forces $q<\frac{n}{n-1}$. For this we break up the integral into the portion near $x$ and the far away part as follows:
$$
\int_{\mathbb{R}^n} |x-y|^{(1-n)q}1_U(y)\ dy = \int_{B(x,2)} |x-y|^{(1-n)q}1_U(y)\, dy + \int_{\mathbb{R}^n\setminus B(x,2)} |x-y|^{(1-n)q}1_U(y)\, dy =I+II.
$$
$I$ you can write as $g*1_U(x)$, where $g(z)=|z|^{(1-n)q}1_{B(0,2)}$, and so by Hölder's inequality we have
$$
|I|\leq \| g\|_{L^t(\mathbb{R}^n)}, \qquad 1<t<\frac{n}{(n-1)q},
$$
where we used that $|U|=1$. You can check by direct computation that this quantity is finite.
To handle $II$, we set $R_k:=\{ y: 2^k\leq |x-y|<2^{k+1}\}$ for $k\geq 1$ and write
$$
II= \sum_{k\geq 1} \int_{R_k} |x-y|^{(1-n)q}1_U(y)\, dy \approx \sum_{k\geq 1} 2^{(1-n)qk} |U\cap R_k|\leq \sum_{k\geq 1} |U\cap R_k|\leq 1.
$$
Combining these the result follows.
We now turn to 2)
This, again by a scaling argument, we can reduce to looking at the unit ball. Here the idea is that $W^{1,n}(B)$ (where $B$ is said unit ball) is contained in every $W^{1,\tilde{p}}(B)$, for $\tilde{p}<n$, and so by the Gagliardo-Nirenberg-Sobolev inequality we have
$$
\| u\|_{L^{\tilde{p}^*}(B)} \lesssim_n \| D u\|_{L^{\tilde{p}}(B)}, \qquad \tilde{p}^*:= \dfrac{n\tilde{p}}{n-\tilde{p}}.
$$
Now simply note that $\tilde{p}^*\to \infty$ as $\tilde{p}\to n^-$.
