Sharp Hessian integrability for the planar Monge--Amp\`ere equation

Abstract

We determine the optimal local integrability of the Hessian for planar convex Alexandrov solutions satisfying $$ \lambda\,dx\leq\mu_u\leq\lambda^{-1}\,dx, \qquad 0<\lambda<1. $$ The distributional Hessian satisfies the sharp endpoint estimate $$ D^2u\in L_{\mathrm{loc}}^{P,\infty}, \qquad P=\frac{1+\lambda}{1-\lambda}. $$ 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: $L^{P,\infty}$ 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.