Sobolev functions on closed subsets of the real line

TL;DR

This paper characterizes how Sobolev functions on the real line behave when restricted to arbitrary closed subsets, using divided differences to provide intrinsic descriptions for different values of p and m.

Contribution

It introduces a new method employing divided differences to intrinsically characterize Sobolev space restrictions on closed subsets of the real line.

Findings

Provides intrinsic characterizations for Sobolev space restrictions.

Applies to all p > 1 and positive integers m.

Uses divided differences as a key tool.

Abstract

For each $p>1$ and each positive integer $m$ we use divided differences to give intrinsic characterizations of the restriction of the Sobolev space $W^m_p(R)$ to an arbitrary closed subset of the real line.