$C^{1,1}$ regularity for degenerate complex Monge-Ampère equations and geodesic rays

Abstract: We prove a $C^{1,1}$ estimate for solutions of complex Monge-Amp`ere equations on compact K\"ahler 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 K\"ahler 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-K\"ahler locus.