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 $$\varepsilon >0$$ , 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.