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Sobolev and Hölder estimates for homotopy operators of the $\overline\partial$-equation on convex domains of finite multitype

Liding Yao

TL;DR

This work develops a universal homotopy operator $\mathcal{H}_q$ for the $\bar{\partial}$-equation on bounded smooth convex domains of finite type, achieving optimal Sobolev, Hölder, Triebel–Lizorkin, and Besov bounds that depend on the domain’s multitype $m_q$. Central to the construction is a Diederich–Fornaess–type integral kernel together with a boundary commutator $[\overline{\partial},\mathcal{E}]$ built from the Rychkov extension, and a decomposition of the kernel into tangential and normal parts controlled via $\varepsilon$-minimal bases. The paper proves weighted kernel estimates and a tangential commutator Hardy–Littlewood framework, enabling $1/m_q$-gain in derivatives and $L^p$–$L^q$ bounds across a wide range of function spaces, including negative indices. It also extends the approach to strongly pseudoconvex domains, yielding improved endpoint regularity, and discusses consequences for the Skew Bergman projection. Overall, the results provide a robust, noncanonical yet highly explicit regularity theory for the $\bar{\partial}$-problem on nontrivial convex domains, with broad applicability in several complex variables.

Abstract

We construct homotopy formulas for the $\overline\partial$-equation on convex domains of finite type that have optimal Sobolev and Hölder estimates. For a bounded smooth finite type convex domain $Ω\subset\mathbb C^n$ that has $q$-type $m_q$ for $1\le q\le n$, our $\overline\partial$ solution operator $\mathcal H_q$ on $(0,q)$-forms has (fractional) Sobolev boundedness $\mathcal H_q:H^{s,p}\to H^{s+1/m_q,p}$ and Hölder-Zygmund boundedness $\mathcal H_q:\mathscr C^s\to\mathscr C^{s+1/m_q}$ for all $s\in\mathbb R$ and $1<p<\infty$. We also show the $L^p$-boundedness $\mathcal H_q:H^{s,p}\to H^{s,pr_q/(r_q-p)}$ for all $s\in\mathbb R$ and $1<p<r_q$, where $r_q:=(n-q+1)m_q+2q$.

Sobolev and Hölder estimates for homotopy operators of the $\overline\partial$-equation on convex domains of finite multitype

TL;DR

This work develops a universal homotopy operator for the -equation on bounded smooth convex domains of finite type, achieving optimal Sobolev, Hölder, Triebel–Lizorkin, and Besov bounds that depend on the domain’s multitype . Central to the construction is a Diederich–Fornaess–type integral kernel together with a boundary commutator built from the Rychkov extension, and a decomposition of the kernel into tangential and normal parts controlled via -minimal bases. The paper proves weighted kernel estimates and a tangential commutator Hardy–Littlewood framework, enabling -gain in derivatives and – bounds across a wide range of function spaces, including negative indices. It also extends the approach to strongly pseudoconvex domains, yielding improved endpoint regularity, and discusses consequences for the Skew Bergman projection. Overall, the results provide a robust, noncanonical yet highly explicit regularity theory for the -problem on nontrivial convex domains, with broad applicability in several complex variables.

Abstract

We construct homotopy formulas for the -equation on convex domains of finite type that have optimal Sobolev and Hölder estimates. For a bounded smooth finite type convex domain that has -type for , our solution operator on -forms has (fractional) Sobolev boundedness and Hölder-Zygmund boundedness for all and . We also show the -boundedness for all and , where .
Paper Structure (9 sections, 28 theorems, 138 equations)

This paper contains 9 sections, 28 theorems, 138 equations.

Table of Contents

  1. Introduction
  2. The Construction of Homotopy Formulas
  3. Estimates via $\varepsilon$-Minimal Bases
  4. Function Spaces and Extension Operators
  5. Tangential Commutator Estimate and Strong Hardy-Littlewood Lemma
  6. Proof of the Theorems
  7. An Additional Result for Smooth Strongly Pseudoconvex Domains
  8. Proof of Proposition \ref{['Prop::LPThm::HLLem']} \ref{['Item::LPThm::HLLem::WtoF']} for $\varepsilon<1$
  9. Boundedness of a Skew Bergman Projection

Key Result

Theorem 1.1

Assume $\Omega\subset\mathbb{C}^n$ is a bounded smooth convex domain of finite type. Then there are operators $\mathcal{H}_q:\mathscr{S}'(\Omega;\wedge^{0,q})\to\mathscr{S}'(\Omega;\wedge^{0,q-1})$ that maps $(0,q)$-forms to $(0,q-1)$-forms with distributional coefficients, for $1\le q\le n$ (we set Moreover suppose $\Omega$ has $q$-type $m_q$ (see Definition Defn::Basis::Type). Then $\mathcal{H}_

Theorems & Definitions (77)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3: Boundedness on Besov spaces
  • Remark 1.4: Boundedness on BMO
  • Lemma 2.1
  • Lemma 2.2
  • Remark 2.3
  • proof
  • Lemma 2.4
  • Definition 2.5
  • ...and 67 more