Sobolev functions on closed subsets of the real line: long version

TL;DR

This paper provides intrinsic characterizations of Sobolev spaces restricted to closed subsets of the real line and demonstrates the near-optimality of the Whitney extension operator for these spaces, with constructive extension criteria.

Contribution

It introduces intrinsic characterizations of Sobolev spaces on closed subsets of the real line and shows the Whitney extension operator is nearly optimal for $L^m_p$ spaces.

Findings

Whitney extension operator is nearly optimal for $L^m_p$-extensions.

Constructive criteria for Sobolev space extensions based on divided differences.

Intrinsic characterizations of Sobolev spaces on arbitrary closed subsets.

Abstract

For each $p>1$ and each positive integer $m$ we give intrinsic characterizations of the restriction of the Sobolev space $W^m_p(R)$ and homogeneous Sobolev space $L^m_p(R)$ to an arbitrary closed subset $E$ of the real line. In particular, we show that the classical one dimensional Whitney extension operator is "universal" for the scale of $L^m_p(R)$ spaces in the following sense: for every $p\in(1,\infty]$ it provides almost optimal $L^m_p$-extensions of functions defined on $E$. The operator norm of this extension operator is bounded by a constant depending only on $m$. This enables us to prove several constructive $W^m_p$- and $L^m_p$-extension criteria expressed in terms of $m^{th}$ order divided differences of functions.