I have trouble with proving the following:
Let $u \in C^{k, \alpha}$. For each $h > 0$, denote the difference quotient of u in the $j$-direction as $\delta_{h, j}u$. That is, \begin{equation} \delta_{h, j} u (x) = \frac{u(x + h e_j) - u(x)}{h}. \end{equation} Suppose $| \delta_{h, j} u |_{k, \alpha} \leq M$ uniformly in $h$. Then $D_j u \in C^{k+1, \alpha}$ and $| D_j u |_{k+1, \alpha} \leq M$.
The Sobolev space version of this argument is easy because we can take a weakly convergent subsequence and pair it with test functions to obtain existence of a higher derivative. However, in the Holder space case, we do not have any test functions to make our lives easy. So I suspect we have to use certain Arzela-Ascoli type compactness, but somehow I am stuck and cannot proceed. I would really appreciate it if somebody could give me some idea or proof. Thank you!