On the Converse of Pansu's Theorem

Guido De Philippis; Andrea Marchese; Andrea Merlo; Andrea Pinamonti; Filip Rindler

doi:10.1007/s00205-024-02059-8

On the Converse of Pansu's Theorem

Arch Ration Mech Anal. 2025;249(1):3. doi: 10.1007/s00205-024-02059-8. Epub 2024 Dec 10.

Authors

Guido De Philippis¹, Andrea Marchese², Andrea Merlo³, Andrea Pinamonti², Filip Rindler⁴

Affiliations

¹ Courant Institute of Mathematical Sciences, New York University, 251 Mercer St., New York, NY 10012 USA.
² Università di Trento, Via Sommarive 14, 38123 Trento, Italy.
³ Universidad del País Vasco (UPV/EHU), Barrio Sarriena S/N 48940, Leioa, Spain.
⁴ Mathematics Institute, University of Warwick, Coventry, CV4 7AL United Kingdom.

Abstract

We provide a suitable generalisation of Pansu's differentiability theorem to general Radon measures on Carnot groups and we show that if Lipschitz maps between Carnot groups are Pansu-differentiable almost everywhere for some Radon measures $μ$ , then $μ$ must be absolutely continuous with respect to the Haar measure of the group.

Keywords: 26A27; 26B05; 28A75; 49Q15; 53C17.