Question in Sobolev space

Question

Let $\Omega$ be a bounded domain and suppose $u\in W^{1,2}(\Omega)\cap C^{0,\alpha}(\Omega)$ for some $0<\alpha<1$. Then is it true that $$u\in W^{1,p}(\Omega)~~\text{ for all } p>1?$$

My attempt: For $1<p<2$ this is true, since $\Omega$ is bounded$\implies$ $W^{1,2}(\Omega)\subset W^{1,p}(\Omega)$. Let $2<p<\infty$. Since $u\in C^{0,\alpha}(\Omega)\subset L^{p}(\Omega)$, for all $p$, we only need to show $|\nabla u|\in L^p(\Omega)$. Is there some inequality to apply? Even any small hints will be helpful.

Answer 1

The function $u(x) = x^\alpha$ belongs to $C^{0,\alpha}(0,1)$. And $u\in W^{1,p}(0,1)$ if and only if $p\in [1,\frac1{1-\alpha})$.