Suppose that $u\colon \mathbb R^d \to \mathbb R$ is twice-differentiable and 1-periodic along each axis. Can the maximum norm of the Hessian be controlled by the maximum norm of the Laplacian, i.e. does there exists $C$ such that $$ \|D^2u\|_{L^{\infty}} \leq \|\Delta u\|_{L^\infty}. $$ I am confused because
- theorems 4.1 and 5.1 of this paper suggest that this follows from standard elliptic regularity theory,
- but this post seems to suggest that the result is not correct.
