Suppose $\{h_{k}\}$ is a sequence of elements in $C^{2,\alpha}$.
I want to show that $\{h_{k}\}$ is uniformly bounded in $C^{2,\alpha}$.
Now, if I'm not mistaken, then $$||h_{k}||_{\displaystyle C^{2,\alpha}} = ||h_{k}||_{\infty} + ||D(h_{k})||_{\infty} + ||D^{2}(h_{k})||_{\infty}+ \sup \left\{ \displaystyle \frac{|h_{k}(x)-h_{k}(y)|}{|x-y|^{\alpha}} \right\} +\\+\sup \left\{ \displaystyle \frac{|D(h_{k}(x))-D(h_{k}(y))|}{|x-y|^{\alpha}} \right\} + \sup \left\{ \displaystyle \frac{|D^{2}(h_{k}(x))-D^{2}(h_{k}(y))|}{|x-y|^{\alpha}} \right\}.$$
So, in order to show that $||h_{k}||_{\displaystyle C^{2,\alpha}} \leq C$, I need to show that each of these summands is $\leq C$.
I'm thinking that the last three terms with $\sup$ in them might be uniformly bounded just by nature of their being $\sup$s. The other ones, I might be able to bound by some other $L_{p}$ norm, if there is any relationship between an $L_{\infty}$ norm and an $L_{2}$ norm, say.
Could somebody please tell me if there is any kind of less than or equal relationship between these norms; i.e., any inequalities that express a relationship between them.