An extremal subharmonic function in non-archimedean potential theory

TL;DR

This paper develops a non-archimedean potential theory analog of the extremal function for subsets of the Berkovich projective line, establishing its relation to Green functions and proving a Brelot-Cartan principle.

Contribution

It introduces a new extremal function in non-archimedean potential theory and proves its equivalence to Green functions for compact sets, along with a Brelot-Cartan principle under certain conditions.

Findings

Extremal function equals the Green function for compact sets.

Established the Brelot-Cartan principle in the non-archimedean setting.

Connected extremal functions with classical potential theory concepts.

Abstract

We define an analog of the Leja-Siciak-Zaharjuta subharmonic extremal function for a proper subset $E$ of the Berkovich projective line $P^1$ over a field with a non-archimedean absolute value, relative to a point $\zeta \not \in E$. When $E$ is a compact set with positive capacity, we prove that the upper semicontinuous regularization of this extremal function equals the Green function of $E$ relative to $\zeta$. As a separate result, we prove the Brelot-Cartan principle, under the additional assumption that the Berkovich topology is second countable.