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 $F(D^{2}u)=0$ 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 $C^{2,\bar{\alpha }(\lambda \mathrm{/}\mathrm{\Lambda })}$ in the interior of the domain, where $0<\lambda \le \mathrm{\Lambda }$ 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 $\tilde{\alpha }=\tilde{\alpha }(\lambda \mathrm{/}\mathrm{\Lambda })>\lambda \mathrm{/}\mathrm{\Lambda }$.