Lusin approximation for horizontal curves in step 2 Carnot groups

TL;DR

This paper proves that all step 2 Carnot groups allow for Lusin approximation of horizontal curves, meaning any such curve can be closely approximated by a smooth one outside a small measure set, extending previous results.

Contribution

It establishes Lusin approximation for horizontal curves in all step 2 Carnot groups, including free groups, and shows this property is preserved under certain Lie group homomorphisms.

Findings

Lusin approximation holds for free step 2 Carnot groups.

The property is preserved under Lie group homomorphisms that preserve the horizontal layer.

All step 2 Carnot groups admit Lusin approximation for horizontal curves.

Abstract

A Carnot group $\mathbb{G}$ admits Lusin approximation for horizontal curves if for any absolutely continuous horizontal curve $\gamma$ in $\mathbb{G}$ and $\varepsilon>0$, there is a $C^1$ horizontal curve $\Gamma$ such that $\Gamma=\gamma$ and $\Gamma'=\gamma'$ outside a set of measure at most $\varepsilon$. We verify this property for free Carnot groups of step 2 and show that it is preserved by images of Lie group homomorphisms preserving the horizontal layer. Consequently, all step 2 Carnot groups admit Lusin approximation for horizontal curves.