On the equivalence of derivatives for maps between Carnot groups

TL;DR

This paper provides an elementary Euclidean-based proof that maps between Carnot groups preserving horizontal curves and having Euclidean horizontal derivatives are also Pansu differentiable, reaffirming key regularity results.

Contribution

It offers a new, simpler proof of Magnani's result on the equivalence of Euclidean and Pansu differentiability for certain Carnot group maps.

Findings

Maps preserving horizontal curves with Euclidean horizontal derivatives are Pansu differentiable.

Reproves Magnani's mean value estimate for Pansu differentiable maps.

Simplifies the proof of a fundamental regularity result in Carnot group theory.

Abstract

This paper gives an alternate, elementary proof of a result of Magnani: maps between Carnot groups that preserve horizontal curves and are continuously differential in horizontal directions in the Euclidean sense are continuously Pansu differentiable. This proof contains primarily Euclidean arguments and also reproves a version of Magnani's mean value estimate for continuously Pansu differentiable maps.