Maximal Plurifinely Plurisubharmonic Functions

The main purpose of this paper is to introduce and study the notion of -maximal -plurisubharmonic functions, which extends the notion of maximal plurisubharmonic functions on a Euclidean domain to an -domain of a", (n) in a natural way. Our main result is that a finite -plurisubharmonic function u on a plurifine domain U satisfies (d d (c) u) (n) = 0 if and only if u is -locally -maximal outside some pluripolar set. In particular, a finite -maximal plurisubharmonic function u satisfies (d d (c) u) (n) = 0.