= Solution
Failure of the bound would give normalized $v_j$ with $\|v_j\|_\infty=1$, zero <Dirichlet boundary condition>, and $\|Lv_j\|_{C^{0,\alpha}}\to0$. The <global Schauder estimate> bounds them in $C^{2,\alpha}$. By <compact embedding of Hölder spaces>, a subsequence converges in $C^{2,\beta}$ to $v$ with $Lv=0$, zero boundary data and $\|v\|_\infty=1$. The uniform <Hölder seminorm> bounds pass to the limit, so $v\in C^{2,\alpha}$. This contradicts the trivial <null space>, proving
$$
\boxed{\|u\|_\infty\leq C_1\|f\|_{C^{0,\alpha}}.}
$$
The <global Schauder estimate> then gives $\|u\|_{C^{2,\alpha}}\leq K\|f\|_{C^{0,\alpha}}$, the bounded-inverse estimate used next.
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