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

The sub-Riemannian Heisenberg group is the simplest nonabelian example of a Carnot group. Suppose $$(\gamma _k)_{0 \le k m}$$ collection continuous function defined on compact set $$K \subset \mathbb {R}$$ taking values in When there horizontal $$C^{m}$$ curve $$\Gamma $$ so that $$D^k \Gamma |_K = \gamma _k$$ for $$k 0, 1,\dots ,m$$ ? Such extensions are known as “Whitney extensions” due to original work Whitney real valued mappings. This question was answered by authors together with Andrea Pinamonti. addition $$\gamma _m$$ uniformly modulus continuity $$\omega . When, then, $$C^{m,\omega }$$ In this paper, we show hypotheses previous $$C^m$$ extension result not sufficient case, and provide new assumptions which necessary guarantee existence such an extension.