Sobolev $W^1_p$-spaces on closed subsets of $R^n$

TL;DR

This paper characterizes the restriction of Sobolev spaces to closed subsets of R^n using local oscillations and doubling measures, providing intrinsic descriptions for p>n.

Contribution

It introduces a new intrinsic characterization of Sobolev space restrictions on closed subsets of R^n for p>n, using local oscillations and doubling measures.

Findings

Provides intrinsic characterizations for Sobolev space restrictions

Uses local oscillations and doubling measures in the analysis

Applicable to arbitrary closed subsets of R^n

Abstract

For each $p>n$ we use local oscillations and doubling measures to give intrinsic characterizations of the restriction of the Sobolev space $W_p^1(R^n)$ to an arbitrary closed subset of $R^n$.