Global Lipschitz extension preserving local constants

TL;DR

This paper introduces a method to extend Lipschitz functions on metric spaces while maintaining local Lipschitz constants, and applies it to prove invariance of Sobolev spaces under metric measure space isomorphisms.

Contribution

It provides a new extension technique that preserves local Lipschitz constants and demonstrates the invariance of Sobolev spaces on metric measure spaces.

Findings

Extension method preserves local Lipschitz constants

Sobolev spaces are invariant under mm-space isomorphisms

Simplifies proofs of invariance results

Abstract

The intent of this short note is to extend real valued Lipschitz functions on metric spaces, while locally preserving the asymptotic Lipschitz constant. We then apply this results to give a simple and direct proof of the fact that Sobolev spaces on metric measure spaces defined with a relaxation approach \`a la Cheeger are invariant under isomorphism class of mm-structures.