Infinite-Dimensional Carnot Groups and Gâteaux Differentiability

  • Enrico Le Donne
  • Sean LiEmail author
  • Terhi Moisala


This paper contributes to the generalization of Rademacher’s differentiability result for Lipschitz functions when the domain is infinite dimensional and has nonabelian group structure. We introduce an infinite-dimensional analogue of Carnot groups that are metric groups equipped with dilations (which we call metric scalable groups) admitting a dense increasing sequence of finite-dimensional Carnot subgroups. For such groups, we show that every Lipschitz function has a point of Gâteaux differentiability. As a step in the proof, we show that a certain \(\sigma \)-ideal of sets that are null with respect to this sequence of subgroups cannot contain open sets. We also give a geometric criterion for when such Carnot subgroups exist in metric scalable groups and provide examples of such groups. The proof of the main theorem follows the work of Aronszajn (Stud Math 57(2):147–190, 1976) and Pansu (Ann Math 129(1):1–60, 1989).


Carnot groups Differentiability Rademacher Gateaux derivative 

Mathematics Subject Classification

28A15 53C17 58C20 46G05 



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Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di PisaPisaItaly
  2. 2.University of ConnecticutMansfieldUSA
  3. 3.Department of Mathematics and StatisticsUniversity of JyväskyläUniversity of JyväskyläFinland

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