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

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 $C^{\mathrm{\infty }}$ Whitney extension result, and a $C^{\mathrm{\infty }}$ 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 $C^{\mathrm{\infty }}$ in the Heisenberg group.