I saw a type of Hölder estimate in Friedman's book: *Partial Differential Equations of Parabolic Type* (page 200, 3.24) which goes as follows:

Suppose we have a uniformly parabolic equation with Hölder coefficients, $$\begin{cases}\partial_tu-Lu=f(x)\quad &\text{in }\Omega\times(0,T]\\ u=0\quad &\text{on }\partial\Omega\\u(x,0)=0\end{cases}$$ where $f(x)=0$ on $\partial \Omega$, then for any $0<\delta<1$, there exist constant $K$ and $\sigma$ such that $$|u|_{C^{1+\delta}(\bar{Q}_T)}\leq KT^\sigma|f|_\infty$$

Right now I am concerned with the version with Neumann boundary condition. Suppose we have a uniformly parabolic equation with Hölder continuous (or as smooth as you want) $$\begin{cases}\partial_tu-Lu=f(x)\quad &\text{in }\Omega\times(0,T]\\\frac{\partial u}{\partial v}=0\quad &\text{on }\partial\Omega\\u(x,0)=0\end{cases}$$ $\Omega$ has smooth boundary and $f$ satisfies the compatibility condition and as smooth as you want, do we have the following Hölder estimate: $$|u|_{C^{1+\delta}(Q_T)}\leq CT^\sigma|f|_{\infty}$$

I believe this is correct, but I don't know where to find the reference. Friedman's book ceased to talk about the Neumann boundary condition case.

Thank you very much.