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

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.