A $C^{m,\omega}$ Whitney Extension Theorem for Horizontal Curves in the Heisenberg Group

TL;DR

This paper provides a characterization of when functions from a compact subset of real numbers into the Heisenberg group can be extended to smooth horizontal curves with controlled regularity, highlighting differences from classical $C^m$ extension conditions.

Contribution

It introduces a $C^{m, ext{omega}}$ Whitney extension theorem for horizontal curves in the Heisenberg group, revealing new conditions necessary for such extensions.

Findings

Characterization of extendability to $C^{m, ext{omega}}$ horizontal curves

Demonstration that direct $C^{m}$-type conditions are insufficient

Identification of the role of the modulus of continuity in extensions

Abstract

We characterize which mappings from a compact subset of $\mathbb{R}$ into the Heisenberg group can be extended to a $C^{m,\omega}$ horizontal curve for a given modulus of continuity $\omega$. We motivate our characterization by showing that the $C^{m,\omega}$ extension property fails if we instead use a more direct analogue of the conditions from the $C^{m}$ case.