Plurifinely Plurisubharmonic functions and the Monge Amp\`ere Operator

TL;DR

This paper extends the Monge-Ampère operator to weakly plurifinely plurisubharmonic functions in complex space, establishing it as a positive measure through new approximation and decomposition techniques.

Contribution

It introduces a novel definition of the Monge-Ampère operator on plurifinely open sets and proves its positivity, using approximation and local decomposition methods.

Findings

Monge-Ampère operator defined on plurifinely plurisubharmonic functions

Proved the operator yields a positive measure

Established approximation of weakly by strongly plurifinely plurisubharmonic functions

Abstract

We will define the Monge-Amp\`ere operator on finite (weakly) plurifinely plurisubharmonic functions in plurifinely open sets in complex n-space and show that it defines a positive measure. Ingredients of the proof include a direct proof for bounded strongly plurifinely plurisubharmonic functions, which is based on the fact that such functions can plurifinely locally be written as difference of ordinary plurisubharmonic functions, and an approximation result stating that weakly plurifinely plurisubharmonic functions are locally limits of strongly finely plurisubharmonic functions in the Dirichlet norm.