C^1,1 regularity for degenerate complex Monge–Ampère equations and geodesic rays

We prove a C^1,1 estimate for solutions of complex Monge–Ampère equations on compact Kähler manifolds with possibly nonempty boundary, in a degenerate cohomology class. This strengthens previous estimates of Phong–Sturm. As applications we deduce the local C^1,1 regularity of geodesic rays in the space of Kähler metrics associated to a test configuration, as well as the local C^1,1 regularity of quasi-psh envelopes in nef and big classes away from the non-Kähler locus.