Extension criteria for homogeneous Sobolev spaces of functions of one variable

Abstract

For each $p>1$ and each positive integer $m$, we give intrinsic characterizations of the restriction of the homogeneous Sobolev space $L_p^m(\mathbb{R})$ 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 $L_p^m(\mathbb{R})$ spaces in the following sense: For every $p\in (1,\infty ]$, it provides almost optimal $L_p^m$-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 $L_p^m$-extension criteria expressed in terms of $m$-th order divided differences of functions.