Let $(M,g)$ be a compact Riemannian manifold and $\Delta_g$ the Laplacian on $M$. Let $L=\Delta_g-\partial_t$ be the heat operator. Let $0<\alpha<1$. Let $u\in C^{2,1}(M\times[0,T])$, $f\in C^0(M\times[0,T])$ such that $f(\cdot,t)\in C^\alpha(M)$ for any $t\in[0,T]$. If $Lu=f$, then for any $t\in[0,T]$, $$\|u(\cdot,t)\|_{C^{2+\alpha}(M)}+\|\partial_tu(\cdot,t)\|_{C^{\alpha/2}(M)}\leq C(M,g,\alpha,T)(\sup_{s\in[0,T]}\|f(\cdot,s)\|_{C^\alpha(M)}+\|u\|_{L^\infty(M\times[0,T])})$$
This is stated on page 182 of the book Variational Problems in Geometry by Seiki Nishikawa, where the author calls it Schauder estimate. However, I cannot deduce this from the standard Schauder estimate, which requires that $f$ is Hölder in $t$ as well.
But this was crucial. In the book a nonlinear parabolic PDE of the form (harmonic map heat flow) $$Lu=\Pi(u)(du,du)$$ where $\Pi(u)(du,du)$ involves only $u$ and first-order $x$-derivatives of $u$, was discussed extensively. An a priori estimate of the form $$\|u(\cdot,t)\|_{C^{2+\alpha}(M)}+\|\partial_tu(\cdot,t)\|_{C^{\alpha/2}(M)}\leq C$$ where $u$ was only assumed to be $C^{2,1}$, was needed. But since we don't know $u$ is Hölder continuous in $t$, the standard Schauder estimate doesn't work here.
So is the above estimate correct? I'm not sure because I haven't yet studied the proof of Schauder estimate. Thanks in advance!
Edit: The constant could depend on $T$.