Lusin approximation for horizontal curves in step 2 Carnot groups

Abstract

A Carnot group $\mathbb{G}$ admits Lusin approximation for horizontal curves if for any absolutely continuous horizontal curve $\gamma$ in $\mathbb{G}$ and $\epsilon >0$, there is a $C^1$ horizontal curve $\Gamma$ such that $\Gamma =\gamma$ and ${\Gamma }^{\prime }={\gamma }^{\prime }$ outside a set of measure at most $\epsilon$. 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.

arXiv comment25 pages