Show that $W^{1,p}(a,b) \subset C^{0,1-\frac{1}{p}}\left([a,b]\right)$

Question

Show that $W^{1,p}(a,b) \subset C^{0,1-\frac{1}{p}}\left([a,b]\right)$ Where the second space are Hölder continuous function with $\gamma = 1-1/p$.

I know that in dimension $n =1$ a function $u$ belonging to $W^{1,p}(a,b)$ is absolutely continuous so I think I should proceed from here and then prove that \begin{equation} |u(y)-u(x)| \le C|x-y|^{1-1/p} \end{equation}

Can you help me?

Answer 1

$u$ is absolutely continuous and $u' \in L^{p}$. So, by Holder's inequality, $|u(x)-u(y)|=|\int_x^{y} u'(t)dt| \leq \|u'\|_p (\int_x^{y} 1^{q}dt)^{1/q}$ where $\frac 1 p +\frac 1 q=1$. Hence the inequality holds with $C=\|u'\|_p $.