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On the Hörmander's estimate

Bingyuan Liu

Abstract

The motivation of the note is to obtain a Hörmander-type $L^2$ estimate for $\bar\partial$ equation. The feature of the new estimate is that the constant is independent of the weight function. Moreover, our estimate can be used for non-plurisubharmonic weight function.

On the Hörmander's estimate

Abstract

The motivation of the note is to obtain a Hörmander-type estimate for equation. The feature of the new estimate is that the constant is independent of the weight function. Moreover, our estimate can be used for non-plurisubharmonic weight function.
Paper Structure (10 sections, 20 theorems, 112 equations)

This paper contains 10 sections, 20 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. The Kähler identity
  4. The Bochner--Kodaira--Nakano identity
  5. Comparison of Laplacians
  6. Estimates involving two equations
  7. Computation involving different weights
  8. The case without boundary (complete metric on $X$)
  9. The case with boundary (incomplete metric on $X$)
  10. Reformulation as a Donnelly--Fefferman type estimate

Key Result

Theorem 1.1

Let $\Omega$ be a bounded pseudoconvex domain in $\mathbb{C}^n$ and $\phi\in\mathrm{PSH}(\Omega)$. Assume that $\phi$ is strictly plurisubharmonic so that for for some $r>0$. Then for the equation $\bar{\partial} u=v$, where $\bar{\partial} v=0$ and $v\in L^2(\Omega, \phi)$, there exists a $u\in L^2(\Omega,\phi)$ so that The $C_0$ is a constant.

Theorems & Definitions (32)

  • Theorem 1.1: Donnelly--Fefferman DF83, see also Chen Ch22
  • Theorem 1.2: Berndtsson--Charpentier BC00, see also Chen Ch22 and Bł ocki Bl13
  • Theorem 1
  • Definition 1.1
  • Proposition 2.1: Demailly De12, Page 270
  • proof
  • Proposition 2.2: Bochner--Kodaira--Nakano Identity, see Page 329 of Demailly De12
  • proof
  • Proposition 2.3
  • proof
  • ...and 22 more