On a remark by Ohsawa related to the Berndtsson-Lempert method for $L^2$-holomorphic extension

TL;DR

This paper advances the Berndtsson-Lempert method for $L^2$-holomorphic extension by constructing a new weight function using Legendre-Fenchel transforms and weak geodesics, confirming a question posed by Ohsawa.

Contribution

It introduces a novel approach to the Berndtsson-Lempert method, providing a new weight function that generalizes previous results and affirms Ohsawa's question.

Findings

Derived a new weight function for the Berndtsson-Lempert method.

Unified previous theorems as special cases of the new result.

Confirmed a question posed by Ohsawa regarding the method.

Abstract

We utilize the Legendre-Fenchel transform and weak geodesics for plurisubharmonic functions to construct a weight function that can be used in the Berndtsson-Lempert method, to give an Ohsawa-Takegoshi extension type of result. Theorem 4.1 and Theorem 0.1 in \cite{OT2017} (Theorem \ref{Theorem A} and \ref{Theorem B} below) follow as two special cases of this result, thus answering affirmatively a question posed by Ohsawa in remark 4.1 in \cite{OT2017}, on the Berndtsson-Lempert method.