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Sobolev and Hölder estimates for the $\overline \partial$ equation on pseudoconvex domains of finite type in $\mathbb C^2$

Ziming Shi

TL;DR

This work advances the global regularity theory for the $\bar{\partial}$-equation on smoothly bounded pseudoconvex domains in $\mathbb{C}^2$ of finite type by constructing almost sharp Sobolev and Hölder–Zygmund estimates via a holomorphic integral formula. The authors develop a holomorphic support-function construction tailored to finite-type geometry, including a nonisotropic polydisc framework and Skoda division to produce Leray data that admit precise, uniformly controlled estimates outside the domain. They then build a homotopy formula with explicit operators $\mathcal{H}_1^{(\epsilon)}$ and $\mathcal{H}_2^{(\epsilon)}$ that map between $H^{s,p}$ and $\Lambda^s$ spaces with a gain of $\frac{1}{m}-\eta$, and prove these bounds extend to a limiting operator on $D$. The approach blends microlocal-like holomorphic division, Range-type exterior-domain reductions, and Rychkov extension techniques to obtain a robust framework that extends classical results of Fefferman–Kohn, Range, and Chang–Nagel–Stein; an outstanding question is whether the small loss $\eta$ can be removed to achieve exact gain $\frac{1}{m}$ in all spaces.

Abstract

We prove a homotopy formula which yields almost sharp estimates in all (positive-indexed) Sobolev and Hölder-Zygmund spaces for the $\overline \partial$ equation on pseudoconvex domains of finite type in $\mathbb C^2$, extending the earlier results of Fefferman-Kohn (1988), Range (1990), and Chang-Nagel-Stein (1992). The main novelty of our proof is the construction of holomorphic support functions that admit precise estimates when the parameter variable lies in a thin shell outside the domain.

Sobolev and Hölder estimates for the $\overline \partial$ equation on pseudoconvex domains of finite type in $\mathbb C^2$

TL;DR

This work advances the global regularity theory for the -equation on smoothly bounded pseudoconvex domains in of finite type by constructing almost sharp Sobolev and Hölder–Zygmund estimates via a holomorphic integral formula. The authors develop a holomorphic support-function construction tailored to finite-type geometry, including a nonisotropic polydisc framework and Skoda division to produce Leray data that admit precise, uniformly controlled estimates outside the domain. They then build a homotopy formula with explicit operators and that map between and spaces with a gain of , and prove these bounds extend to a limiting operator on . The approach blends microlocal-like holomorphic division, Range-type exterior-domain reductions, and Rychkov extension techniques to obtain a robust framework that extends classical results of Fefferman–Kohn, Range, and Chang–Nagel–Stein; an outstanding question is whether the small loss can be removed to achieve exact gain in all spaces.

Abstract

We prove a homotopy formula which yields almost sharp estimates in all (positive-indexed) Sobolev and Hölder-Zygmund spaces for the equation on pseudoconvex domains of finite type in , extending the earlier results of Fefferman-Kohn (1988), Range (1990), and Chang-Nagel-Stein (1992). The main novelty of our proof is the construction of holomorphic support functions that admit precise estimates when the parameter variable lies in a thin shell outside the domain.

Paper Structure

This paper contains 4 sections, 34 theorems, 207 equations.

Table of Contents

  1. Introduction
  2. Function spaces
  3. Construction of the holomorphic support function
  4. Estimates for the $\overline\partial$ homotopy operator

Key Result

Theorem 1.1

Let $D \subset \mathbb{C}^2$ be a ($C^\infty$-)smoothly bounded pseudoconvex domain of finite type $m$. For each $\eta>0$, there exist linear operators $\mathcal{H}^\eta_i$, $i=1,2$ such that Here $H^{s,p}(D)$ is the fractional Sobolev space (see Definition Def::Sobolev_Dom), and $\Lambda^s(D)$ is the Hölder-Zygmund space (see Definition Def::H-Z). $H^{s,p}_{(0,i)}(D)$ ($i=1,2$) denotes the space

Theorems & Definitions (69)

  • Theorem 1.1
  • Definition 2.1: Hölder-Zygmund space on $\mathbb{R}^N$
  • Definition 2.2: Hölder-Zygmund space on domains
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5: Sobolev space on $\mathbb{R}^N$
  • Definition 2.6: Sobolev space on domains
  • Remark 2.7
  • Definition 2.8
  • Definition 2.9
  • ...and 59 more