  The goal of this work is to prove the regularity of certain
quasi-plurisubharmonic upper envelopes. Such envelopes appear in a natural way
in the construction of hermitian metrics with minimal singularities on a big
line bundle over a compact complex manifold. We prove that the complex Hessian
forms of these envelopes are locally bounded outside an analytic set of
singularities. It is furthermore shown that a parametrized version of this
result yields a priori inequalities for the solution of the Dirichlet problem
for a degenerate Monge-Ampere operator; applications to geodesics in the space
of Kahler metrics are discussed. A similar technique provides a logarithmic
modulus of continuity for Tsuji's "supercanonical" metrics, which generalize a
well-known construction of Narasimhan-Simha.
