Reference request: Parabolic Schauder estimates for the heat equation with $f \in L^\infty$

Question

Let us consider the heat equation $$\partial_t u - \Delta u = f(x, t) \quad \text{in }Q_R $$ where $Q_R = B_R \times (-R^2,0].$ I would like to know the kind of regularity we should expect of $u$ if we assume $f \in L^\infty(Q_R)$.

I know that, for $f \in C^{\alpha}(Q_R)$, then by the parabolic Schauder estimate, $u \in C^{2 + \alpha, 1 + \alpha/2}(Q_R)$ and $$\|u \|_{C^{2 + \alpha, 1 + \alpha/2}(Q_{R/2})} \le C\left(\frac{1}{R^{2 + \alpha}} |u|_{0;Q_R} + \frac{1}{R^{\alpha}} |f|_{0;Q_R} + [f]_{\alpha, \alpha/2;Q_R}\right).$$ Moreover, in the elliptic case $\Delta u = f$ in $B_1$, if we assume $f$ only bounded $f \in L^\infty$, then $u \in C^{1,1 - \varepsilon}$ inside $B_1$ for any $\varepsilon \in (0,1)$ and satisfies $$\|u\|_{C^{1,1 - \varepsilon}(B_{1/2})} \le C_\varepsilon \left(\|f\|_{L^\infty(B_{1})} + \|u\|_{L^\infty(B_{1})}\right).$$ Therefore, I would be expecting the same type of behaviour for the parabolic case, meaning that $u \in C^{1 + \alpha, \alpha/2}(Q_{R/2})$ with the appropriate estimate. However, I have been looking for a reference proving such result, but I couldn't find it anywhere. Any help would be appreciated.

Answer 1

The estimates $$\|u\|_{C^{1,1 - \varepsilon}(B_{1/2})} \le C_\varepsilon \left(\|f\|_{L^\infty(B_{1})} + \|u\|_{L^\infty(B_{1})}\right)$$ is not a Schauder estimate. In fact, it comes form an interior $L^p$ estimates (Calderon-Zygmund estimates) for $-\Delta u=f$ and the Sobolev embedding. That is,
$$ \|u\|_{W^{2,p}(B_1)} \leq C \|u\|_{L^p(B_2)} +C\|f\|_{L^p(B_2)} \leq C \|u\|_{L^{\infty}(B_2)} +C\|f\|_{L^{\infty}(B_2)}.$$ And $$\|u\|_{C^{1,1-\epsilon}(B_1)}\leq C \|u\|_{W^{2,\frac{n}{\epsilon}}(B_1)}.$$ As for the heat equation, the results is similar to the above.