Maximal Plurifinely Plurisubharmonic Functions

The main purpose of this paper is to introduce and study the notion of -maximal F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {F}$\end{document}-plurisubharmonic functions, which extends the notion of maximal plurisubharmonic functions on a Euclidean domain to an -domain of ℂn in a natural way. Our main result is that a finite -plurisubharmonic function u on a plurifine domain U satisfies (ddcu)n= 0 if and only if u is -locally -maximal outside some pluripolar set. In particular, a finite -maximal plurisubharmonic function u satisfies (ddcu)n = 0.