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New Estimates of Rychkov's Universal Extension Operator for Lipschitz Domains and Some Applications

Ziming Shi, Liding Yao

Abstract

Given a bounded Lipschitz domain $Ω\subset\mathbb R^n$, Rychkov showed that there is a linear extension operator $\mathcal E$ for $Ω$ which is bounded in Besov and Triebel-Lizorkin spaces. In this paper we introduce some new estimates for the extension operator $\mathcal E$ and give some applications. We prove the equivalent norms $\|f\|_{\mathscr A_{pq}^s(Ω)}\approx\sum_{|α|\le m}\|\partial^αf\|_{\mathscr A_{pq}^{s-m}(Ω)}$ for general Besov and Triebel-Lizorkin spaces. We also derive some quantitative smoothing estimates of the extended function and all its derivatives on $\overlineΩ^c$ up to the boundary.

New Estimates of Rychkov's Universal Extension Operator for Lipschitz Domains and Some Applications

Abstract

Given a bounded Lipschitz domain , Rychkov showed that there is a linear extension operator for which is bounded in Besov and Triebel-Lizorkin spaces. In this paper we introduce some new estimates for the extension operator and give some applications. We prove the equivalent norms for general Besov and Triebel-Lizorkin spaces. We also derive some quantitative smoothing estimates of the extended function and all its derivatives on up to the boundary.

Paper Structure

This paper contains 20 sections, 24 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. The general results for Rychkov-type operators
  3. An application to several complex variables
  4. Terminology and organization
  5. Historical Remarks
  6. Some previous extension operators
  7. Two approaches to equivalent norms on domains
  8. Boundedness On Besov and Triebel-Lizorkin Spaces
  9. Preliminary and preparations
  10. Proof of Theorem \ref{['Thm::MainThm']} \ref{['Item::MainThm::Bdd']}
  11. The zeroth term of $E_\omega$
  12. The Anti-derivatives: Proof of Theorem \ref{['Thm::MainThm']} \ref{['Item::MainThm::AntiDev']}
  13. The Quantitative Smoothing Estimate: Proof of Theorem \ref{['Thm::MainThm']} \ref{['Item::MainThm::Smoothing']}
  14. Sketch of the proof
  15. The computations
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $\Omega\subset\mathbb{R}^n$ be a bounded Lipschitz domain. Let $0<p,q\le\infty$ ($p<\infty$ for the $\mathscr{F}$-cases) and $s\in\mathbb{R}$. For any positive integer $m$, there is a $C=C_{\Omega,p,q,s,m}>0$ such that

Theorems & Definitions (74)

  • Theorem 1.1: Equivalent norms in Lipschitz domains
  • Theorem 1.2
  • Remark 1.3
  • Definition 1.4
  • Theorem 1.5
  • Remark 1.6
  • Proposition 1.7: Anti-derivatives with support constraint
  • Remark 1.8
  • Definition 1.9
  • Definition 1.11
  • ...and 64 more