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