On the optimal regularity of weak geodesics in the space of metrics on a polarized manifold

Let (X,L) be a polarized compact manifold, i.e. L is an ample line bundle over X and denote by H the infinite dimensional space of all positively curved Hermitian metrics on L equipped with the Mabuchi metric. In this short note we show, using Bedford-Taylor type envelope techniques developed in the authors previous work [ber2], that Chen's weak geodesic connecting any two elements in H are C^{1,1}-smooth, i.e. the real Hessian is bounded, for any fixed time t, thus improving the original bound on the Laplacins due to Chen. This also gives a partial generalization of Blocki's refinement of Chen's regularity result. More generally, a regularity result for complex Monge-Ampere equations over X\timesD, for D a pseudconvex domain in \C^{n} is given.