On the sharp Hessian integrability conjecture in the plane

We prove that if $u\in C^0\left(B_1\right)$ satisfies $F(x,D^{2}u)\le 0$ in $B_1\subset R^2$, in the viscosity sense, for some fully nonlinear $(\lambda ,\Lambda )$-elliptic operator, then $u\in W^{2,\epsilon }\left(B_{1/2}\right)$, with appropriate estimates, for a sharp exponent $\epsilon =\epsilon (\lambda ,\Lambda )$ verifying$\frac{1.629}{\frac{\Lambda }{\lambda }+1}<\epsilon (\lambda ,\Lambda )\le \frac{2}{\frac{\Lambda }{\lambda }+1}.$ The upper bound is conjectured to be the optimal one. Thus, the main new information proven in this paper is that the sharp Hessian integrability exponent for viscosity supersolutions in the plane remains at least 81.45% of its upper bound. This greatly improves previous known estimates.