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The existence of a bounded linear extension operator for $L^{s,p}(\mathbb{R}^n)$ when $\frac{n}{p}<\{s\}$

Han Li

Abstract

Let $L^{s,p}(\mathbb{R}^n)$ denote the homogeneous Sobolev-Slobodeckij space. In this paper, we demonstrate the existence of a bounded linear extension operator from the jet space $J^{\lfloor s \rfloor}_E L^{s,p}(\mathbb{R}^n)$ to $L^{s,p}(\mathbb{R}^n)$ for any $E \subseteq \mathbb{R}^n$, $p \in [1, \infty)$, and $s \in (0, \infty)$ satisfying $\frac{n}{p} < \{s\}$, where $\{s\}$ represents the fractional part of $s$. Our approach builds upon the classical Whitney extension operator and uses the method of exponentially decreasing paths.

The existence of a bounded linear extension operator for $L^{s,p}(\mathbb{R}^n)$ when $\frac{n}{p}<\{s\}$

Abstract

Let denote the homogeneous Sobolev-Slobodeckij space. In this paper, we demonstrate the existence of a bounded linear extension operator from the jet space to for any , , and satisfying , where represents the fractional part of . Our approach builds upon the classical Whitney extension operator and uses the method of exponentially decreasing paths.

Paper Structure

This paper contains 5 sections, 12 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Main Lemmas
  4. Proof of main result Theorem \ref{['main']}
  5. Proof of Lemma \ref{['10']}

Key Result

Theorem 1

Let $n \in \mathbb{N}^*$, $s \in (0, \infty)$, and $p \in [1, \infty)$. If $\frac{n}{p} < \{s\}$, where $\{s\}$ is the fractional part of $s$, and $E \subseteq \mathbb{R}^n$, then there exists a bounded linear extension operator $T : J^{\lfloor s \rfloor}_E L^{s,p}(\mathbb{R}^n) \to L^{s,p}(\mathbb{ where $C_{n,s,p}$ is a positive constant depending only on $n$, $s$, and $p$.

Theorems & Definitions (25)

  • Theorem 1: Main Result
  • Remark 1
  • Definition 1
  • Definition 2: Homogeneous Hölder spaces
  • Proposition 1: Sobolev Embedding Theorem Leoni2023A
  • Proposition 2
  • proof
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • ...and 15 more