 Necessity of Hölder forcing for Schauder estimates

The bounded-forcing norm alone cannot control the Hölder seminorm of the Hessian matrix. On a unit Euclidean ball take $u_m=m^{-2}\sin (m{x}_1)$ and $f_m=\Delta u_m=-\sin (m{x}_1)$. The derivative supremum norm $|u_m{|}_2$ and $\Vert f_m{\Vert }_{\infty }$ stay bounded, but the Hessian matrix difference between zero and $\pi /(2m)e_1$ gives $[D^2{u}_m{]}_{\alpha }\ge (2m/\pi {)}^{\alpha }$ on $B_{1/2}$.