A well-known theorem of Evans and Krylov states that in an equation of the form $F(D^2 u)=g$, provided that the right hand side and $u$ both have Lipschitz gradient, and that $F$ is concave or convex and of class $C^2$ on the space of symmetric matrices, and uniformly elliptic at $u$, the solution is of class $C^{2,\alpha}$ for some $\alpha$.

The proof in Gilbarg-Trudinger uses differentiaion of the equation twice, and therefore $L^\infty$ second derivatives of $g$ are needed. My question is about minimal regularity known to be necessary for the function $g$. Does one really need Lipscitz gradients for $g$ or it can be relaxed?