Doubt in Moorey's ineqality given in Sobolev space .

Question

I am going through the statement of Moorey's inequality given in chapter $5$ of evans pde book on page $266$.

Statement is: Assume $p > n$. Then there exist a constant $C$, depending on $p, n$ such that $$\|u\|_{C^{0, \gamma}(\mathbb{R}^n)} \leq C \|u\|_{W^{1,p}(\mathbb{R}^n)}$$ for all $u \in C^1(\mathbb{R}^n) $ where $\gamma = 1-\frac{n}{p}$.

Now i am consfused in the following

$1)$ How can we say that any function in $C^1(\mathbb{R}^n) $ is also in $C^{0, \gamma}(\mathbb{R}^n)$.

$2)$ How any function in $C^1(\mathbb{R}^n) $ is also in $W^{1,p}(\mathbb{R}^n) $

I am confused in this statement as for $(1)$ to hold $u$ should have compact support. For (2) to hold is that $u$ and $Du$ (derivative) should be in $L^p(\mathbb{R}^n)$. Am i right or wrong?

Answer 1

The theorem states that if $u\in C^1(\mathbb{R}^n)$ is no element of $C^{0,\gamma}(\mathbb{R}^n)$(which might be the case) it is not in $W^{1,p}(\mathbb{R}^n)$ (under the given circumstances)So

$||u||_{C^{0,\gamma}(\mathbb{R}^n)}=\infty$ implies $||u||_{W^{1,p}(\mathbb{R}^n)}=\infty$(this for (1)). As for (2) You are right again: $C^1(\mathbb{R}^n)$ is not contained in $W^{1,p}(\mathbb{R}^n)$ but then the right hand side of the inequality is $\infty$ and the inequality still holds in a sense. The only point You are wrong : Hölder continuous functions need not have compact support (take a constant function as an example).(See the next Theorem in Evans for an application of Theorem 4)