A note on maximal plurifinely plurisubharmonic functions

TL;DR

This paper investigates the properties of plurifinely locally maximal plurifinely plurisubharmonic functions, establishing conditions under which these functions are maximal and clarifying their relationship with finite plurisubharmonic functions.

Contribution

It improves existing results by proving that locally bounded plurifinely locally maximal functions are maximal and characterizing maximality for finite plurisubharmonic functions.

Findings

Locally bounded plurifinely locally maximal functions are maximal.

Finite plurisubharmonic functions are maximal iff they are plurifinely locally maximal.

Abstract

In this note we study the plurifinely locally maximal plurifinely plurisubharmonic functions and improve some known results on these functions. We prove in particular that any locally bounded plurifinely locally maximal plurifinely plurisubharmonic function is maximal, and that any finite plurisubharmonic function on an open subset of $\mathbb{C}^n$ is maximal if and only if it is plurifinely locally maximal.