Sobolev, BV and perimeter extensions in metric measure spaces

TL;DR

This paper investigates the conditions under which sets and functions can be extended in metric measure spaces, establishing equivalences for BV and perimeter properties and exploring Sobolev space extendability with illustrative examples.

Contribution

It characterizes the strong BV extension property in terms of perimeter set extensions and analyzes relationships between BV, perimeter, and Sobolev space extendability in metric spaces.

Findings

Open set has strong BV extension iff it has strong perimeter extension

Implications between BV extension and Sobolev space extendability are partially established

Examples show some implications between properties do not hold universally

Abstract

We study extensions of sets and functions in general metric measure spaces. We show that an open set has the strong BV extension property if and only if it has the strong extension property for sets of finite perimeter. We also prove several implications between the strong BV extension property and extendability of two different non-equivalent versions of Sobolev $W^{1,1}$-spaces and show via examples that the remaining implications fail.