A lifting theorem for operators on spaces of Lipschitz functions

TL;DR

The paper establishes a lifting theorem for bounded linear operators on Lipschitz function spaces, showing they can be extended along the De Leeuw embedding with controlled norms.

Contribution

It proves that every bounded linear operator between Lipschitz spaces admits a near-isometric lifting along the De Leeuw embedding.

Findings

Operators can be lifted with arbitrarily small norm increase.

Lifting preserves the structure of Lipschitz functions.

The result applies to pointed metric spaces.

Abstract

We prove that every bounded linear operator between Lipschitz spaces admits a lifting along the De Leeuw embedding. More precisely, given pointed metric spaces $M$ and $N$ and $\epsilon>0$, every bounded linear operator $S:\mathrm{Lip}_0(M)\to \mathrm{Lip}_0(N)$ admits a lifting $\mathfrak{S}:C(\beta \tilde{M})\to C(\beta \tilde{N})$ such that $\|\mathfrak{S}\|\leq \|S\|+\epsilon$ and $\mathfrak{S}(\varPhi_M(f))=\varPhi_N(S(f))$ for every $f\in \mathrm{Lip}_0(M)$.