Calculus of Variations and Geometric Measure Theory

M. Capolli - A. Pinamonti - G. Speight

A $C^m$ Lusin Approximation Theorem for Horizontal Curves in the Heisenberg Group

created by pinamonti on 20 Aug 2019
modified on 27 Jul 2021


Accepted Paper

Inserted: 20 aug 2019
Last Updated: 27 jul 2021

Journal: Calc. Var. Partial Differential Equations
Year: 2021


We prove a $C^m$ Lusin approximation theorem for horizontal curves in the Heisenberg group. This states that every absolutely continuous horizontal curve whose horizontal velocity is $m-1$ times $L^1$ differentiable almost everywhere coincides with a $C^m$ horizontal curve except on a set of small measure. Conversely, we show that the result no longer holds if $L^1$ differentiability is replaced by approximate differentiability. This shows our result is optimal and highlights differences between the Heisenberg and Euclidean settings.