$C^{1,1}$ regularity for degenerate complex Monge-Amp\`ere equations and geodesic rays

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.