Sharp Hessian integrability for the planar Monge--Ampère equation

We determine the optimal local integrability of the Hessian for planar convex Alexandrov solutions satisfying $$\lambda dx\le {\mu }_u\le {\lambda }^{-1}dx,0<\lambda <1.$$ The distributional Hessian satisfies the sharp endpoint estimate $$D^{2}u\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^{P,\mathrm{\infty }},P=\frac{1+\lambda }{1-\lambda }\mathrm{.}$$ This settles the weak estimate proposed by De Philippis and Figalli and identifies the larger optimal exponent $P$. We further construct strictly convex $C^1$ solutions for which the distribution function of the Laplacian is comparable to $T^{-P}$ for all sufficiently large $T$. Hence both the exponent and the weak Lorentz endpoint are optimal: $L^{P,\mathrm{\infty }}$ cannot be replaced by $L^{P,b}$ for any finite $b>0$. As applications, we obtain sharp estimates in Minty coordinates and optimal quantitative bi-Lipschitz restrictions for gradient maps, including an optimal exceptional-mass bound in quadratic-cost transport.