Vector spaces of delta-plurisubharmonic functions and extensions of the complex Monge-Ampère operator (Q465191)

The spaces of interest in this paper are the spaces of functions which can be represented as the difference of two negative plurisubharmonic functions or more specifically some cones of negative plurisubharmonic functions (Cegrell classes).NEWLINENEWLINEThe paper begins with some instructive preliminaries on Riesz spaces and the theory of the Monge-Ampère equation. Next, one finds a discussion on the radially symmetric plurisubharmonic functions and the differences of them. After that the authors propose two extensions of the Monge-Ampère operator which are shown to be distinct by an example. The Dirichlet problem with respect to these functions is studied and also some uniqueness questions are considered.NEWLINENEWLINEAs a side effect the paper establishes a link between the theory of ordered vector spaces that are not locally convex, with pluripotential theory.NEWLINENEWLINEOne of the main tools of this paper is the abstract Kantorovich extension theorem on Riesz spaces.NEWLINENEWLINEThe paper contains several open problems in the last section.