How to show this Holder bound? Define the seminorm on the space $S=[0,1]\times[0,T]$
$$\mid u\mid_{\alpha} = \sup\frac{|u(x, t) - u(y,s)|}{(|x-y|^2 + |t-s|)^{\frac{\alpha}{2}}}.$$
Define the norms on the same space
$$\lVert u \rVert_{C^{0, \alpha}} = \lVert u \rVert_{C^0} + \mid u\mid_{\alpha}$$
and
$$\lVert u \rVert_{C^{2, \alpha}} = \lVert u \rVert_{C^0} +\lVert u_x \rVert_{C^0}+\lVert u_{xx} \rVert_{C^0}+\lVert u_t \rVert_{C^0}+ \mid u_{xx}\mid_{\alpha} + \mid u_t\mid_{\alpha}.$$
Suppose that $\lVert u \rVert_{C^2, \alpha} \leq C$ where $C$ is a constant. Let $a, b, c \in C^{0, \alpha}$. How can I show that
$$\lVert au_{xx} + bu_x + cu\rVert_{C^{0, \alpha}} \leq K\lVert u \rVert_{C^{2, \alpha}}$$
for some constant $K$?
Or equivalently, want to show that
$$\sup_{\lVert u \rVert_{C^{2,\alpha}} \leq C_1}\lVert au_{xx} + bu_x + cu\rVert_{C^{0, \alpha}} \leq K_1$$
(ALL the above norms are over the compact set $S$).
Thanks for any help
 A: If $f,g\in C^{\alpha}(S)$, then for all $(x,t),(x',t')\in S$, we have 
\begin{align}
|f\cdot g(x,t)-f\cdot g(x',t')|&=|f(x,t)(g(x,t)-g(x',t'))+g(x',t')f(x,t)-f(x',t')g(x',t')|\\\
&\leq \lVert f\rVert_{\infty}|g(x,t)-g(x',t')|+\lVert f\rVert_{\infty}|f(x,t)-f(x',t')|,
\end{align}
hence 
\begin{equation}\[f\cdot g\]_{\alpha}\leq \lVert f\rVert_{\infty}\[g\]_{\alpha}+\lVert g\rVert_{\infty} \[f\]_{\alpha}.
\end{equation}
We deduce that 
\begin{align}
\lVert au_{xx}+bu_x+cu\rVert_{C^{0,\alpha}}&\leq \max(\lVert a\rVert_{\infty},\lVert b\rVert_{\infty},\lVert c\rVert_{\infty})(\lVert u_{xx}\rVert_{\infty}+\lVert u_x\rVert_{\infty}+\lVert u\rVert_{\infty})\\\
&+\max(\lVert a\rVert_{\infty},\lVert b\rVert_{\infty},\lVert c\rVert_{\infty})(\[ u_{xx}\]_{\alpha}+\[u_x\]_{\alpha}+\[u\]_{\alpha}) \\\
&+\max(\[a\]_{\alpha},\[b\]_{\alpha},\[c\]_{\alpha})(\lVert u_{xx}\rVert_{\infty}+\lVert u_x\rVert_{\infty}+\lVert u\rVert_{\infty})\\\
&\leq \max(\max(\[a\]_{\alpha},\[b\]_{\alpha},\[c\]_{\alpha}),\max(\lVert a\rVert_{\infty},\lVert b\rVert_{\infty},\lVert c\rVert_{\infty}))\lVert u\rVert_{C^{2,\alpha}}.
\end{align}
We can get an equivalent norm on $C^{2,\alpha}(S)$ defining 
\begin{equation}\lVert u\rVert:=\[u\]_{\alpha}+\lVert u\rVert_{\infty}+\[u_t\]_{\alpha}+\lVert u_t\rVert_{\infty}+\[u_x\]_{\alpha}+\lVert u_x\rVert_{\infty}+\[u_{xx}\]_{\alpha}+\lVert u_{xx}\rVert_{\infty}.
\end{equation}
This can be shown using mean value theorem. 
