Extension criteria for homogeneous Sobolev space of functions of one variable

TL;DR

This paper provides intrinsic criteria for extending functions from arbitrary closed subsets of the real line to the entire line within homogeneous Sobolev spaces, using a universal Whitney extension operator with near-optimal bounds.

Contribution

It characterizes restrictions of homogeneous Sobolev spaces to closed sets and demonstrates the universality and near-optimality of the Whitney extension operator for these spaces.

Findings

Provides intrinsic characterization of Sobolev space restrictions

Shows Whitney extension operator is nearly optimal for all p

Establishes constructive extension criteria based on divided differences

Abstract

For each $p>1$ and each positive integer $m$ we give intrinsic characterizations of the restriction of the homogeneous Sobolev space $L^m_p(R)$ to an arbitrary closed subset $E$ of the real line. 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 $L^m_p$-extension criteria expressed in terms of $m^{th}$ order divided differences of functions.