# A solution of an $L^{2}$ extension problem with optimal estimate and applications

@article{Guan2013ASO, title={A solution of an \$L^\{2\}\$ extension problem with optimal estimate and applications}, author={Qi’an Guan and Xiangyu Zhou}, journal={arXiv: Complex Variables}, year={2013} }

In this paper, we prove an $L^2$ extension theorem with optimal estimate in a precise way, which implies optimal estimate versions of various well-known $L^2$ extension theorems. As applications, we give proofs of a conjecture of Suita on the equality condition in Suita's conjecture, the so-called $L-$conjecture, and the extended Suita conjecture. As other applications, we give affirmative answer to a question by Ohsawa about limiting case for the extension operators between the weighted…

## 129 Citations

ON THE OPTIMAL
$L^{2}$
EXTENSION THEOREM AND A QUESTION OF OHSAWA

- MathematicsNagoya Mathematical Journal
- 2020

Abstract In this paper, we present a version of Guan-Zhou’s optimal
$L^{2}$
extension theorem and its application. As a main application, we show that under a natural condition, the question posed…

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- 2018

We prove a general version of Siu’s lemma for plurisubharmonic functions with nontrivial multiplier ideal sheaves and use it to prove an optimal $$L^2$$L2 extension theorem and an optimal…

Equality in Suita's conjecture

- Mathematics
- 2018

Without using the $L^2$ extension theorem, we provide a new proof of the equality part in Suita's conjecture, which states that for any open Riemann surface admitting a Green function, the Bergman…

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- Mathematics
- 2015

We present the theory of twisted $$L^2$$L2 estimates for the Cauchy–Riemann operator and give a number of recent applications of these estimates. Among the applications: extension theorem of…

A simplified proof of optimal $L^2$-extension theorem and extensions from non-reduced subvarieties.

- Mathematics
- 2019

We give a simplified proof of an optimal version of the Ohsawa-Takegoshi $L^2$-extension theorem. We follow the variational proof by Berndtsson-Lempert and use the method in the paper of…

Weighted
$$L^2$$
L
2
Approximation of Analytic Sections

- MathematicsThe Journal of Geometric Analysis
- 2022

In this paper, we obtain a global weighted $$L^2$$ L 2 approximation result for holomorphic sections in weighted Bergman spaces, generalizing the approximation theorems of Taylor, Fornæss, and Wu.…

A Survey on L 2 Extension Problem

- Mathematics
- 2015

In the present paper, we’ll give a survey of our recent results on the L2 extension problem with optimal estimate. We’ll consider the problem in various settings according to Ohsawa’s series papers,…

Potential Theoretic Hyperbolicity and $\mathbf{L^2}$ extension. Part I: Stein manifolds

- Mathematics
- 2014

We establish a new generalization of an $L^2$ extension theorem of Ohsawa-Takegoshi type. The improvement in the theorem is that it allows the usual curvature assumptions to be significantly weakened…

On sharper estimates of Ohsawa–Takegoshi $L^2$-extension theorem

- MathematicsJournal of the Mathematical Society of Japan
- 2019

We present an $L^2$-extension theorem with an estimate depending on the weight functions for domains in $\mathbb{C}$. This sharpens known optimal estimates. In radial weight cases, we prove that our…

$L^2$ estimates and existence theorems for the $\bar{\partial}$ operators in infinite dimensions, I

- Mathematics
- 2020

The classical $L^2$ estimate for the $\overline{\partial}$ operators is a basic tool in complex analysis of several variables. Naturally, it is expected to extend this estimate to infinite…

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