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