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On $1/2$ estimate for global Newlander-Nirenberg theorem

Ziming Shi

Abstract

Given a formally integrable almost complex structure $X$ defined on the closure of a bounded domain $D \subset \mathbb C^n$, and provided that $X$ is sufficiently close to the standard complex structure, the global Newlander-Nirenberg problem asks whether there exists a global diffeomorphism defined on $\overline D$ that transforms $X$ into the standard complex structure, under certain geometric and regularity assumptions on $D$. In this paper we prove a quantitative result of this problem. Assuming $D$ is a strictly pseudoconvex domain in $\mathbb C^n$ with $C^2$ boundary, and that the almost structure $X$ is of the Hölder-Zygmund class $Λ^r(\overline D)$ for $r>\frac{3}{2}$, we prove the existence of a global diffeomorphism (independent of $r$) in the class $Λ^{r+\frac12-\varepsilon}(\overline D)$, for any $\varepsilon>0$.

On $1/2$ estimate for global Newlander-Nirenberg theorem

Abstract

Given a formally integrable almost complex structure defined on the closure of a bounded domain , and provided that is sufficiently close to the standard complex structure, the global Newlander-Nirenberg problem asks whether there exists a global diffeomorphism defined on that transforms into the standard complex structure, under certain geometric and regularity assumptions on . In this paper we prove a quantitative result of this problem. Assuming is a strictly pseudoconvex domain in with boundary, and that the almost structure is of the Hölder-Zygmund class for , we prove the existence of a global diffeomorphism (independent of ) in the class , for any .
Paper Structure (7 sections, 20 theorems, 150 equations)

This paper contains 7 sections, 20 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Function spaces
  4. Smoothing operator on bounded Lipschitz domains
  5. Remarks on stability of constants
  6. Transformation of perturbation under diffeomorphism
  7. Iteration scheme and convergence of maps

Key Result

Theorem 1.1

Let $D$ be a domain in $\mathbb{C}^n$ with $C^2$ boundary that is strictly pseudoconvex with respect to the standard complex structure in $\mathbb{C}^n$, for $n \geq 2$. Let $3/2 < m \leq \infty$ and let $\{ X_{\overline \alpha} = \frac{\partial }{\partial \overline z_\alpha} + A_{\overline \alpha}^

Theorems & Definitions (35)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2: Hölder-Zygmund space
  • Lemma 2.3
  • Lemma 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Proposition 2.9
  • ...and 25 more