Higher order Whitney extension and Lusin approximation for Horizontal curves in the Heisenberg group

TL;DR

This paper establishes advanced Whitney extension and Lusin approximation results for horizontal curves in the Heisenberg group, covering various smoothness classes and completing the theoretical framework in this area.

Contribution

It proves new $C^{m, ext{omega}}$ finiteness, Lusin approximation, and Whitney extension theorems for horizontal curves in the Heisenberg group, extending previous work to broader smoothness classes.

Findings

Proves $C^{m, ext{omega}}$ finiteness principle.

Establishes $C^{m, ext{omega}}$ Lusin approximation.

Provides $C^{ ext{infty}}$ Whitney extension and Lusin approximation results.

Abstract

In the setting of horizontal curves in the Heisenberg group, we prove a $C^{m,\omega}$ finiteness principle, a $C^{m,\omega}$ Lusin approximation result, a $C^{\infty}$ Whitney extension result, and a $C^{\infty}$ Lusin approximation result. Combined with previous work, this completes the study of Whitney extension and Lusin approximation for horizontal curves of class $C^{m}$, $C^{m,\omega}$, and $C^{\infty}$ in the Heisenberg group.