Isometric composition operators on Lipschitz spaces

TL;DR

This paper characterizes when composition operators induced by Lipschitz maps are isometric between Lipschitz spaces, answering a question posed by Weaver, under the condition that the space X has the peak property.

Contribution

It provides a complete characterization of basepoint-preserving Lipschitz maps inducing isometric composition operators on Lipschitz spaces with the peak property.

Findings

Characterization of isometric composition operators on Lipschitz spaces.

Answer to Weaver's open question.

Conditions involving the peak property of X.

Abstract

Given pointed metric spaces $X$ and $Y$, we characterize the basepoint-preserving Lipschitz maps $\phi$ from $Y$ to $X$ inducing an isometric composition operator $C_\phi$ between the Lipschitz spaces $Lip_0(X)$ and $Lip_0(Y)$, whenever $X$ enjoys the peak property. This gives an answer to a question posed by N. Weaver in his book [Lipschitz algebras. Second edition. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2018].