Bi-Lipschitz invariance of planar $BV$- and $W^{1,1}$-extension domains

TL;DR

This paper proves that bi-Lipschitz transformations preserve the property of being a $BV$- or $W^{1,1}$-extension domain in the plane, establishing invariance under such mappings.

Contribution

It demonstrates the bi-Lipschitz invariance of extension domain properties for planar $BV$ and $W^{1,1}$ spaces, a previously unestablished geometric invariance.

Findings

Bi-Lipschitz images of planar $BV$-extension domains remain $BV$-extension domains.

Bi-Lipschitz images of planar $W^{1,1}$-extension domains remain $W^{1,1}$-extension domains.

Abstract

We prove that a bi-Lipschitz image of a planar $BV$-extension domain is also a $BV$-extension domain, and that a bi-Lipschitz image of a planar $W^{1,1}$-extension domain is again a $W^{1,1}$-extension domain.