Sharper estimates of Ohsawa--Takegoshi $L^2$-extension theorem in higher dimensional case

TL;DR

This paper extends sharper $L^2$-extension estimates of the Ohsawa--Takegoshi theorem to higher dimensions, utilizing pluricomplex Green functions and Azukawa pseudometric, advancing the understanding of optimal extension constants.

Contribution

It establishes higher dimensional sharper estimates of the Ohsawa--Takegoshi theorem using pluricomplex Green functions and introduces an analogue of Berndtsson--Lempert type $L^2$-extension theorem.

Findings

Higher dimensional sharper $L^2$-extension estimates are proven.

An analogue of Berndtsson--Lempert type theorem is established.

The results include $L^2$-minimum extension in the radial case.

Abstract

Hosono obtained sharper estimates of the Ohsawa--Takegoshi $L^2$-extention theorem by allowing the constant depending on the weight function for a domain in $\mathbb{C}$. In this article, we show the higher dimensional case of sharper estimates of the Ohsawa--Takegoshi $L^2$-extention theorem. To prove the higher dimensional case of them, we establish an analogue of Berndtsson--Lempert type $L^2$-extension theorem by using the pluricomplex Green functions with poles along subvarieties. As a special case, we consider the sharper estimates in terms of the Azukawa pseudometric and show that the higher dimensional case of sharper estimate provides the $L^2$-minimum extension for radial case.