Hessian estimates for Lagrangian mean curvature equation with sharp Lipschitz phase

We establish a prior interior $C^{1,1}$ estimates for convex solutions and supercritical phase solutions to the Lagrangian mean curvature equation with sharp Lipschitz phase. Counter-examples exist when the phase is Hölder continuous but not Lipschitz. As an application we obtain interior $C^{2,\alpha }$ regularity for $C^0$ viscosity solutions on the first phase interval $((n-2)\frac{\pi }{2},n\frac{\pi }{2})$.