On the space of Kahler potentials

We consider the geodesic equation for the generalized Kahler potential with only mixed second derivatives bounded. We show that given such two generalized Kahler potentials, there is a unique geodesic segment such that for each point on the geodesic, the generalized Kahler potential has uniformly bounded mixed second derivatives (in manifold directions). This generalizes a fundamental theorem of Chen \cite{Chen00} on the space of Kahler potentials.