Hölder estimate of solution to linear parabolic PDE

Question

Consider a standard Cauchy-Dirichlet problem for linear parabolic PDE: \begin{cases} u_t + \mathcal{L}u=-g, \ (t,x) \in [0,T)\times \Omega \\ u(T,x)=0 , \ x\in \Omega\\ u|_{\partial \Omega}=0 \end{cases} Under nice assumptions on $\mathcal{L}$, $g$ and $\partial \Omega$, we have the well known (boundary) Schauder estimate: $$\exists C>0, \ |u|_{2+ \alpha} \leq C|g|_{\alpha},$$ where for $0<\alpha<1$, $|g|_{\alpha}$ denotes the $\alpha$-Hölder norm of $g$: $$|g|_{\alpha} := \sup_{(t,x)}|g(t,x)| + \sup_{\substack{(t,x)\\(\tilde{t},\tilde{x})}}\dfrac{|g(t,x)-g(\tilde{t},\tilde{x})|}{\sqrt{|x-\tilde{x}|^2 + |t-\tilde{t}|}^{\ \alpha}}, $$ and $$|u|_{2+\alpha}:=|u|_{\alpha}+\sum|D_x u|_{\alpha}+\sum|D_x ^2 u|_{\alpha}+|D_t u|_{\alpha}. $$ The above estimate essential says that the $(2+\alpha)$ norm of the solution can be bounded by the $\alpha$-norm of $g$. However in my current application, I only need a bound on the $\alpha$-norm of the solution, but in terms of the infinity norm of $g$, i.e. something like: $$|u|_{\alpha} \leq C \cdot \sup_{(t,x)}|g(t,x)|.$$ My question is: is there a standard reference to this type of Hölder estimate of solution to linear parabolic PDEs? Thank you very much!