Calculus of Variations and Geometric Measure Theory
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E. Le Donne - G. Speight

Lusin Approximation for Horizontal Curves in Step 2 Carnot Groups

created by speight on 08 Feb 2016
modified on 16 May 2017


Published Paper

Inserted: 8 feb 2016
Last Updated: 16 may 2017

Journal: Calculus of Variations and Partial Differential Equations
Volume: 55
Number: 5
Pages: 1-22
Year: 2016


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.


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