On extending $L^{2}$ holomorphic functions from complex hyperplanes

TL;DR

This paper demonstrates that the curvature term in the Kohn-Morrey-Hörmander inequality suffices for $ar{ ext{d}}$-estimates in the extension of $L^{2}$ holomorphic functions from hyperplanes, simplifying the proof of the Ohsawa-Takegoshi theorem.

Contribution

It shows that the curvature term alone can produce the necessary $ar{ ext{d}}$-estimate, utilizing the self boundedness of weight gradients to modify weights.

Findings

Curvature term sufficiency for $ar{ ext{d}}$-estimates

Use of self boundedness of weight gradients

Simplification of Ohsawa-Takegoshi proof

Abstract

The key to the proof of the Ohsawa-Takegoshi Extension Theorem is a certain $\bar{\partial}$-estimate. The purpose of this note is to show that the 'curvature term' that arises in the Kohn-Morrey-H\"{o}rmander inequality (or the Bochner-Kodaira technique) is sufficient to produce such an estimate. We exploit self boundedness of the gradients of the weight functions to change the weight with respect to which the adjoint is taken. The weights, on the other hand, are the usual ones used in this context.