A new plurisubharmonic capacity and functions holomorphic along holomorphic vector fields

TL;DR

This paper introduces a generalized capacity theory extending Siciak's projective capacity, and proves a broader version of Forelli's theorem for functions holomorphic along specific vector fields, advancing complex analysis.

Contribution

It develops a new plurisubharmonic capacity and extends Forelli's theorem to functions holomorphic along integral curves of diagonalizable vector fields.

Findings

Generalized capacity theory based on plurisubharmonic functions

Extended Forelli's theorem to aligned vector fields

Improved results over previous theorems

Abstract

The main purpose of this article is to present a generalization of Forelli's theorem for functions holomorphic along a suspension of integral curves of a diagonalizable vector field of aligned type. For this purpose, we develop a new capacity theory that generalizes the theory of projective capacity introduced by Siciak \cite{Siciak82}. Our main theorem improves the results of \cite{KPS09}, \cite{Cho22} as well as the original Forelli's theorem.