Yang Zhou

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

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}....