  In sub-Riemannian geometry there exist, in general, no known explicit
representations of the heat kernels, and these functions fail to have any
symmetry whatsoever. In particular, they are not a function of the control
distance, nor they are for instance spherically symmetric in any of the layers
of the Lie algebra. Despite these unfavourable aspects, in this paper we
establish a new heat semigroup characterisation of the Sobolev and $BV$ spaces
in a Carnot group by means of an integral decoupling property of the heat
kernel.
