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Sobolev extension in a simple case

Marjorie Drake, Charles Fefferman, Kevin Ren, Anna Skorobogatova

Abstract

In this paper, we establish the existence of a bounded, linear extension operator $T: L^{2,p}(E) \to L^{2,p}(\mathbb{R}^2)$ when $1<p<2$ and $E$ is a finite subset of $\mathbb{R}^2$ contained in a line.

Sobolev extension in a simple case

Abstract

In this paper, we establish the existence of a bounded, linear extension operator when and is a finite subset of contained in a line.
Paper Structure (11 sections, 22 theorems, 71 equations, 3 figures)

This paper contains 11 sections, 22 theorems, 71 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Main Result
  3. Constants
  4. Dyadic Squares and Intervals
  5. Some Elementary Inequalities
  6. The Set $E$
  7. The Calderón-Zygmund Decomposition
  8. Groups of Shadows and Squares
  9. Assigning special points to CZ squares
  10. The Norm of an Interpolant
  11. Removing the extra point from $E^+$

Key Result

Theorem 1

There exists a linear map $T:L^{2,p}(E)\to L^{2,p}({\mathbb R}^2)$ such that $Tf=f$ on $E$ and $\Vert Tf \Vert_{L^{2,p}({\mathbb R}^2)}\leq C_p \Vert f \Vert_{L^{2,p}(E)}$ for every $f\in L^{2,p}(E)$. Moreover, we can take $T$ independent of $p$, such that for each point $z \in {\mathbb R}^2$, the v

Figures (3)

  • Figure 1: Two possible scenarios for a CZ square $Q$.
  • Figure 2: CZ squares $Q_1,Q_2,Q_3\in \mathop{\mathrm{GP}}\nolimits(z^*)$ for a point $z^*\in E$.
  • Figure 3: Two possible scenarios for a contactless child $Q'$ of $Q^{++}$.

Theorems & Definitions (36)

  • Theorem 1
  • Theorem 2
  • Remark 1.1
  • Theorem 3
  • Corollary 5.1
  • Remark 5.1
  • Corollary 5.2
  • Corollary 5.3
  • Corollary 5.4
  • Lemma 5.5
  • ...and 26 more