  We obtain $C^{2,\beta}$ estimates up to the boundary for solutions to
degenerate Monge-Amp\`ere equations of the type $$ \det D^2 u =
f~~\text{in}~\Omega, \quad \quad ~f\sim \text{dist}^{\alpha}(\cdot,
\partial\Omega)~\text{near}~\partial\Omega,~\alpha>0. $$ As a consequence we
obtain global $C^\infty$ estimates up to the boundary for the eigenfunctions of
the Monge-Amp\`ere operator $(\det D^2 u)^{1/n}$ on smooth, bounded, uniformly
convex domains in $R^n$.
