An Ohsawa-Takegoshi-type $L^2$ extension for upper semi-continuous $L^2$-optimal functions

TL;DR

This paper extends the Ohsawa-Takegoshi $L^2$-extension theorem to upper semi-continuous $L^2$-optimal functions, providing new characterizations of plurisubharmonic functions and verifying key properties like Skoda's theorem.

Contribution

It introduces an $L^2$-extension theorem for upper semi-continuous $L^2$-optimal functions and explores their properties, including characterization of plurisubharmonic functions and validation of Skoda's theorem.

Findings

Characterization of plurisubharmonic functions via $L^2$-estimate properties

Verification that upper semi-continuous $L^2$-optimal functions satisfy Skoda's integrability theorem

Proof that these functions exhibit the strong openness property

Abstract

In this article, we obtain an Ohsawa-Takegoshi-type $L^2$-extension for upper semi-continuous $L^2$-optimal functions via a Lebesgue-type differentiation theorem. As applications, we give a characterization of plurisubharmonic functions via the multiple coarse $L^2$-estimate property for (strongly) upper semi-continuous functions and show that (strongly) upper semi-continuous $L^2$-optimal functions satisfy Skoda's integrability theorem and the strong openness property.