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



How to show this Holder bound?
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. 


