Necessity of Hölder forcing for Schauder estimates
= 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 $u_m=m^{-2}\sin(mx_1)$ and $f_m=\Delta u_m=-\sin(mx_1)$. The <derivative supremum norm> $|u_m|_2$ and $\|f_m\|_\infty$ stay bounded, but the <Hessian matrix> difference between zero and $\pi/(2m)e_1$ gives $[D^2u_m]_\alpha\geq(2m/\pi)^\alpha$ on $B_{1/2}$.