# Extension criteria for homogeneous Sobolev spaces of functions of one variable

### Pavel Shvartsman

Technion - Israel Institute of Technology, Haifa, Israel

## Abstract

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

## Cite this article

Pavel Shvartsman, Extension criteria for homogeneous Sobolev spaces of functions of one variable. Rev. Mat. Iberoam. 37 (2021), no. 1, pp. 361–414

DOI 10.4171/RMI/1210