Nonexistence of linear operators extending Lipschitz (pseudo)metric

TL;DR

This paper demonstrates that there exist specific zero-dimensional compact metric spaces where no continuous linear extension operator can extend Lipschitz pseudometrics from a subspace to the entire space, highlighting limitations in extension theory.

Contribution

It provides a counterexample showing the nonexistence of linear extension operators for Lipschitz pseudometrics in certain zero-dimensional compact spaces.

Findings

Counterexample of nonexistence of linear extension operators

Based on Brudnyi and Brudnyi's results on Lipschitz function extensions

Highlights limitations in Lipschitz pseudometric extension theory

Abstract

We present an example of a zero-dimensional compact metric space $X$ and its closed subspace $A$ such that there is no continuous linear extension operator for the Lipschitz pseudometrics on $A$ to the Lipschitz pseudometrics on $X$. The construction is based on results of A. Brudnyi and Yu. Brudnyi concerning linear extension operators for Lipschitz functions.