The Whitney extension theorem is a classical result in analysis giving a
necessary and sufficient condition for a function defined on a closed set to
be extendable to the whole space with a given class of regularity. It has
been adapted to several settings, including the one of Carnot groups.
However, the target space has generally been assumed to be equal to
for
some
.
We focus here on the extendability problem for general ordered pairs
(with
nonabelian). We analyse
in particular the case
and
characterize the groups
for which the Whitney extension property holds, in terms of a newly introduced
notion that we call
pliability. Pliability happens to be related to rigidity as defined by
Bryant and Hsu. We exploit this relation in order to provide examples of
nonpliable Carnot groups, that is, Carnot groups such that the Whitney
extension property does not hold. We use geometric control theory results on the
accessibility of control affine systems in order to test the pliability of a Carnot
group. In particular, we recover some recent results by Le Donne, Speight
and Zimmerman about Lusin approximation in Carnot groups of step 2
and Whitney extension in Heisenberg groups. We extend such results to all
pliable Carnot groups, and we show that the latter may be of arbitrarily large
step.
