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

TL;DR

This paper proves a $C^m$ Lusin approximation theorem for horizontal curves in the Heisenberg group, establishing optimal conditions under which such curves can be approximated by smooth ones, and highlighting differences from Euclidean cases.

Contribution

The paper introduces a $C^m$ Lusin approximation theorem for horizontal curves in the Heisenberg group, demonstrating optimality and contrasting with Euclidean settings.

Findings

The theorem holds for curves with $L^1$ differentiable horizontal velocity.

The result fails if replaced by approximate differentiability.

Highlights differences between Heisenberg and Euclidean geometries.

Abstract

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.