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On curve-flat Lipschitz functions and their linearizations

Gonzalo Flores, Mingu Jung, Gilles Lancien, Colin Petitjean, Antonín Procházka, Andrés Quilis

TL;DR

This work establishes a sharp bridge between metric-geometric properties of Lipschitz maps and Banach-space operator ideals via the Lipschitz-free functor. The central result characterizes when the linearization $\widehat{f}$ of a Lipschitz map is Dunford-Pettis or Radon-Nikodým by a curve-flatness condition on $f$, and shows these properties are equivalent to not fixing any copy of $L_1$; the paper also connects these operator-theoretic properties to strong RN and representability in special cases. The authors prove that the DP and RN properties are compactly determined and provide a compact-reduction framework to lift these results beyond compact domain spaces; they also study factorization questions through $p$-1-u spaces and give concrete counterexamples illustrating the limitations of certain natural extensions. Collectively, the results extend AGPP’s Banach-space characterizations to the level of Lipschitz maps and their linearizations, enriching the interaction between metric geometry and operator ideals with concrete, applicable criteria and counterexamples. The work also highlights several open directions, including factorization through Schur/RNP spaces and the precise scope of strong RN for non-compact domains, offering a roadmap for further synthesis of metric and linear theories.

Abstract

We show that several operator ideals coincide when intersected with the class of linearizations of Lipschitz maps. In particular, we show that the linearization $\widehat{f}$ of a Lipschitz map $f:M\to N$ is Dunford-Pettis if and only if it is Radon-Nikodým if and only if it does not fix any copy of $L_1$. We also identify and study the corresponding metric property of $f$, which is a natural extension of the curve-flatness.

On curve-flat Lipschitz functions and their linearizations

TL;DR

This work establishes a sharp bridge between metric-geometric properties of Lipschitz maps and Banach-space operator ideals via the Lipschitz-free functor. The central result characterizes when the linearization of a Lipschitz map is Dunford-Pettis or Radon-Nikodým by a curve-flatness condition on , and shows these properties are equivalent to not fixing any copy of ; the paper also connects these operator-theoretic properties to strong RN and representability in special cases. The authors prove that the DP and RN properties are compactly determined and provide a compact-reduction framework to lift these results beyond compact domain spaces; they also study factorization questions through -1-u spaces and give concrete counterexamples illustrating the limitations of certain natural extensions. Collectively, the results extend AGPP’s Banach-space characterizations to the level of Lipschitz maps and their linearizations, enriching the interaction between metric geometry and operator ideals with concrete, applicable criteria and counterexamples. The work also highlights several open directions, including factorization through Schur/RNP spaces and the precise scope of strong RN for non-compact domains, offering a roadmap for further synthesis of metric and linear theories.

Abstract

We show that several operator ideals coincide when intersected with the class of linearizations of Lipschitz maps. In particular, we show that the linearization of a Lipschitz map is Dunford-Pettis if and only if it is Radon-Nikodým if and only if it does not fix any copy of . We also identify and study the corresponding metric property of , which is a natural extension of the curve-flatness.
Paper Structure (24 sections, 33 theorems, 75 equations, 1 figure)

This paper contains 24 sections, 33 theorems, 75 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lipschitz-free spaces
  4. Preliminaries on operators and operator ideals
  5. Preliminaries on curve-flat Lipschitz functions
  6. Preliminaries on martingales
  7. General preliminaries
  8. Dunford-Pettis Lipschitz operators
  9. f is curve-flat implies f hat is Dunford-Pettis: The compact case
  10. Compact determination of Dunford-Pettis operators
  11. Proof of Theorem \ref{['t:DP']}
  12. Radon-Nikodým Lipschitz operators
  13. Proof of the compact case of Theorem \ref{['t:RN']}
  14. Compact determination of RN operators
  15. Proof of Theorem \ref{['t:RN']}
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $M$ and $N$ be pointed metric spaces and assume that $M$ is complete. For a 0-preserving Lipschitz map $f:M\to N$, the following are equivalent:

Figures (1)

  • Figure 1: The construction of the factorization is best understood on a picture.

Theorems & Definitions (70)

  • Theorem 1.1
  • Theorem : Area Formula, Kirchheim
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Corollary 2.4
  • proof
  • Lemma 2.5
  • ...and 60 more