Regularity for critical points of convex functionals on Hessian spaces

We consider variational integrals of the form $int F(D^2u)$ where $F$ is convex and smooth on the Hessian space. We show that a critical point $uin W^{2,infty}$ of such a functional under compactly supported variations is smooth if the Hessian of $u$ has a small oscillation.