Local Hessian integrability for planar Monge--Ampère inequalities

Let \(0<\lambda<1\) and let \(u\) be a convex Alexandrov solution on a bounded convex planar domain, with Monge--Ampère density between \(\lambda\) and \(\lambda^{-1}\). We prove that its full distributional Hessian belongs locally to strong \(L^{p_\lambda}\), where \(p_\lambda=(1+\lambda^2)/(1-\lambda^2)\). This implies the corresponding weak endpoint estimate posed in the Monge--Ampère regularity literature. The proof uses a change of variables with a Lipschitz inverse and a local weighted scalar Lipschitz projection. An energy estimate on the set where the projection differs from the original map controls the cutoff error and yields the required negative moment.