Extension criteria for homogeneous Sobolev space 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 $L_p^m(R)$ 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 $L_p^m(R)$ 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.

48 pages. arXiv admin note: substantial text overlap with arXiv:1808.01467, arXiv:1710.07826