I wish to prove that if $kp>n$, $k-\frac{n}{p}\notin\mathbb{N}$, then we have the embedding $W^{k,p}(\Omega)\hookrightarrow C^{l,\alpha}(\Omega)$. I assume that i already have the case for $k=1$ already established, i.e. $W^{1,p}\hookrightarrow C^{0,\alpha}$. For simplicity sake, say I wish to show that $W^{2,p}\hookrightarrow C^{1,\alpha}$. Suppose $u\in W^{2,p}$, so that $\nabla u\in W^{1,p}$. By the base case I have that $\nabla u \in C^{0,\alpha}$. But $\nabla u$ is a priori only a weak gradient which is (Hoelder) continuous. How can I conclude from here that $\nabla u$ is indeed a strong gradient, and hence conclude that $u\in C^{1,\alpha}$?

Update: I was suggested the following method. For a (1-d sobolev function) we have the fundamental theorem of calculus: $u(x)-u(a)=\int_a^xu'(y)dy$. Then certainly, if u' is continuous, u is differentiable. Now we would like to say that this works for all directional derivatives for a sobolev function in dimension $n$. However, the restriction of a sobolev function to a subspace may not be a sobolev function, and so this method seems to fail. E.g. 1/|x| is square integrable in $\mathbb{R}^2$, but not square integrable when we restrict it to the line.


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