  This paper is devoted to the regularity analysis of a geodesic equation in
the space of Sasakian metrics. Firstly, we reduce the geodesic equation in the
space of Sasakian metrics to a Dirichlet problem of degenerate complex
Monge-Amp\'ere type eqution on the K\"ahler cone; secondly, we obtain a priori
etimates for the above equation. These a priori estimates guarantee the
existence and uniqueness of $C^{2}_{w}$ geodesic for any two points in the
space of Sasakian metrics. We also give some geometric applications of the
above estimates in the end of this paper.
