Plurisubharmonic and holomorphic functions relative to the plurifine topology

TL;DR

This paper introduces weak and strong notions of plurifinely plurisubharmonic and holomorphic functions within the plurifine topology, exploring their properties, regularization, and stability under composition, advancing pluripotential theory.

Contribution

It defines and studies the weak concept of plurifinely plurisubharmonic functions, establishing their characterization, regularization, and behavior under composition with holomorphic maps.

Findings

Weak plurifinely plurisubharmonic functions are characterized by composition with affine-linear maps.

Regularization of pointwise supremum preserves weak plurifine plurisubharmonicity.

Weak plurifine properties are stable under composition with weakly plurifinely holomorphic maps.

Abstract

A weak and a strong concept of plurifinely plurisubharmonic and plurifinely holomorphic functions are introduced. Strong will imply weak. The weak concept is studied further. A function f is weakly plurifinely plurisubharmonic if and only if f o h is finely subharmonic for all complex affine-linear maps h. As a consequence, the regularization in the plurifine topology of a pointwise supremum of such functions is weakly plurifinely plurisubharmonic, and it differs from the pointwise supremum at most on a pluripolar set. Weak plurifine plurisubharmonicity and weak plurifine holomorphy are preserved under composition with weakly plurifinely holomorphic maps.