On the smoothness of solutions of fully nonlinear second order equations in the plane

We study interior $C^{2,\alpha}$ regularity estimates for solutions of fully nonlinear uniformly elliptic equations of the general form $F(D^2u)=0$ in two independent variables and without any geometric condition on $F$. By means of the theory of divergence form equations we prove that $C^2$ solutions of the previous equation are $C^{2,\bar\alpha(\lambda/\Lambda)}$ in the interior of the domain, where $0<\lambda\leq\Lambda$ are the ellipticity constants. We finally exploit the theory of nondivergence equations in the plane to obtain $C^{2,\tilde\alpha}$ regularity for an explicit exponent $\tilde\alpha=\tilde\alpha(\lambda/\Lambda)>\lambda/\Lambda$.