On the Geodesics in the space of K\"ahler metrics with prescribed singularities

Motivated by the results of B. Berndtsson, in this memoir we use the new estimates developed by W. He to extend a theorem of the second author on the existence of weak $C^{1,1}$ geodesics between two smooth non-degenerate K\"ahler potentials to the case where the metrics on the end points may have singularities on some analytic set and may be positive semi-definite.