Setting: Let $(M,g)$ be a compact Riemannian manifold without boundary. Let $\Delta\_g$ be the Laplacian and $L=\Delta\_g-\partial\_t$ the heat operator. Let $0<\alpha<1$, $0<t\_0<T$.


Let $$u\in C^{2,1}(M\times[0,T))\cap C^\infty(M\times(0,T))$$ solve $$Lu=f$$ where $$f\in C^0(M\times[0,T))\cap C^\infty(M\times(0,T))$$ is such that $f(\cdot,t)\in C^\alpha(M)$ for any $t\in[0,T)$ and $$\sup\_{t\in[0,T)}\|f(\cdot,t)\|\_{C^\alpha(M)}<\infty.$$



> 
> QUESTION: Do we have the follow *a priori* estimate?
>  $$\sup\_{t\in[t\_0,T)}\|u(\cdot,t)\|\_{C^{2+\alpha}(M)}\leq C\bigg(\sup\_{s\in[t\_0,T]}\|f(\cdot,s)\|\_{C^\alpha(M)}+\|u\|\_{L^\infty(M\times[0,T])}\bigg)$$
> 
> 
> 


This estimate was needed for proving the global existence of a solution for the harmonic map heat flow, which is a quasilinear parabolic equation: $$(\Delta\_g-\partial\_t)u=\Pi(u)(du,du).$$ Basically it says that $u(\cdot,t)$ does not blow up as $t\nearrow T$.


This is stated on page 182 of the book *Variational Problems in Geometry* by Seiki Nishikawa, where the author calls it Schauder estimate. However, the standard Schauder estimate, which involves the Hölder seminorm of $f$ in $t$ as well. The author lists Gilbarg–Trudinger (which obviously do not talk about parabolic equations) and a Japanese book (I don't speak Japanese) for such Schauder estimates.


So is the above estimate correct? If so, where can I find a proof?


Thanks in advance!


Edit: A possible approach might be to use the heat kernel, so that $u$ has an inegral expression which we can analyze. However I'm totally unfamiliar with estimates on the heat kernel. For example, is the estimate true for $M=\mathbb{R}^n$?


Edit: On page 316 of *Partial Differential Equations III: Nonlinear Equations* (Second Edition) by Michael E. Taylor, $(1.13)$ is the following $$\|e^{t\Delta}\|\_{\mathcal{L}(C^r,C^{r+s})}\leq C\_st^{-s/2},\quad0<t\leq1,$$ where $\Delta$ is the Laplacian on a complete Riemannian manifold. This implies my wanted estimate, but unfortunately no proof was given. I skimmed the earlier parts of the book and think maybe this follows from the estimates of pseudo-differential operators acting on Hölder–Zygmund spaces that is established earlier. Am I right?



Reference request: Schauder estimate in the space variable for parabolic equations
For the heat equation in $\mathbb R^n$ the estimate holds. The main thing is the smoothness of the volume potential $g=\Gamma\*f$, where $\Gamma$ is the fundamental solution. It is obtained in O. A. Ladyzenskaja, V. A. Solonnikov, and N. N. Ural'ceva, *Linear and Quasi-linear Equations of Parabolic Type*, ch.4, $\S2$. Though in the beginning of the paragraph the assumption on $f$ is Holder continuity wrt $x$ and $t$ both, in the process it is proven that 
$$
|D^2\_xg(x,t)-D^2\_xg(x',t)|\le C|x-x'|\sup\_{s\in[0,T]}\|f(\cdot,s)\|\_{C^\alpha(\mathbb R^n)},\quad x,x'\in \mathbb R^n,\ t>0.
$$


