I see that there are a couple of topics on this question, but neither of them has the answer I am looking for.
In general the goal is to prove the Morrey's inequality $\|u\|_{C^{0,\gamma}(\mathbb{R}^n)}\leq C\|u\|_{W^{1,p}(\mathbb{R}^n)}$, $\gamma=1-\frac{n}{p}$ which is equivalent to $\sup_{{\mathbb{R}}^n}|u|+\sup_{x\neq y}\left\{ \frac{|u(x)-u(y)|}{|x-y|^{1-\frac{n}{p}}}\right\}\leq C\|u\|_{W^{1,p}(\mathbb{R}^n)}$. And the only part I don't understand is the following:
The proof comes to the conclusions that:
(1) $\sup_{{\mathbb{R}}^n}|u|\leq C\|u\|_{W^{1,p}(\mathbb{R}^n)}$
and
(2) $[u]_{C^{0,1-\frac{n}{p}}(\mathbb{R}^n)}=\sup_{x\neq y}\left\{ \frac{|u(x)-u(y)|}{|x-y|^{1-\frac{n}{p}}}\right\}\leq C\|Du\|_{{L^p}(\mathbb{R}^n)}$.
And then it says that (1) and (2) lead to the desired result. I assume that (1) and (2) are summed, but in this case, we would obtain
$\sup_{{\mathbb{R}}^n}|u|+\sup_{x\neq y}\left\{ \frac{|u(x)-u(y)|}{|x-y|^{1-\frac{n}{p}}}\right\}\leq C\|u\|_{W^{1,p}(\mathbb{R}^n)}+C\|Du\|_{{L^p}(\mathbb{R}^n)}$, which is not what the Morrey's inequality is.
Could anyone please give me a hint! Thank you.