I asked this question on math.stackexchange.com, however I didn't get any answers so I'll try it here.


Let $M$ be a compact manifold without boundary. Consider $Lu:=\partial\_tu-\Delta u$. Let $f\in C^{0,0,\alpha}((0,T)\times M,\mathbb{R})$, $u\_0\in C^{2,\alpha}(M,\mathbb{R})$ and let $u\in C^{1,2,\alpha}((0,T)\times M,\mathbb{R})$ be the unique solution of


$$
\begin{cases} 
Lu=f \\
 u(t,.)\to u\_0(.) & \text{as $t\to 0$}
\end{cases}
$$


Here $C^{k,l,\alpha}$, $C^{k,\alpha}$ are the appropriate (parabolic) Hölder spaces.


Then we have 


$$||u||\_{C^{1,2,\alpha}((0,T)\times M,\mathbb{R})}\le C(||f||\_{C^{0,0,\alpha}((0,T)\times M,\mathbb{R})}+||u\_0||\_{C^{2,\alpha}(M,\mathbb{R})})$$


For some $C>0$ independent of $f,$ $u$ and $u\_0$.



> 
> **Question**: Is it possible to choose $C$ independent of $T$? If C depends on $T$, i.e. $C=C(T)$, is it possible to choose $C(T)$ in a way that $C(T)$ is bounded for $T\to 0$?
> 
> 
> 


If one replaces $M$ by a suitable domain in $\mathbb{R}^n$ the answer of my question seems to be "Yes". However, I need the result for compact manifolds and was wondering if there occur any problems.



Schauder estimate for the heat equation on compact manifolds
The constant $C(T)$ depends on $T$, consider $f=1$, $u\_0=0$ and the solution $u=t$. But it is bounded for $T\to0$. The rhs $f\in C^{0,\alpha}([0,T]\times M)$ can be continued to $\tilde f\in C^{0,0,\alpha}([0,1]\times M)$, $||\tilde f||\_{C^{0,0,\alpha}((0,1)\times M)}= |f||\_{C^{0,0,\alpha}((0,T)\times M)}$. Denote $\tilde u$ the solution with rhs $\tilde f$. Then
$$
|u||\_{C^{2,1,\alpha}((0,T)\times M)}\le |\tilde u||\_{C^{2,1,\alpha}((0,1)\times M)}\le
$$
$$
\le C(1)(||\tilde f||\_{C^{0,0,\alpha}((0,1)\times M)}+||u\_0||\_{C^{2,\alpha}(M,\mathbb{R})}))\le C(1) (|f||\_{C^{0,0,\alpha}((0,T)\times M)}+||u\_0||\_{C^{2,\alpha}(M,\mathbb{R})})).
$$


