I saw this in a proof I am reading, and have been unable to justify the statement. Assume $\Omega$ to be a open bounded set in $R^n$ with smooth boundary, and $u\in C^{2,\alpha}(\bar\Omega)$. Then,for some $a,b\in R$,
$\|u\|_{C^\alpha(\Omega)}\leq a\|u\|_{C^{2,\alpha}(\Omega)}+b\|u\|_{C^0(\Omega)}$
Does anyone have a hint for this ?
It smells like Young's inequality to me, but I can't do the full proof.
$\|u\|_{C^\alpha(\Omega)}=\|u\|_{C^0}+[u]_{\alpha}$, where $[u]_{\alpha}=\sup_{x\neq y} \frac{\|u(x)-u(y)\|}{\|x-y\|^{\alpha}}$
$\|u\|_{C^{2,\alpha}(\Omega)}=\|u\|_{C^2}+[D^2u]_{\alpha}$