  We consider a heat-type operator L structured on the left invariant
1-homogeneous vector fields which are generators of a Carnot group, multiplied
by a uniformly positive matrix of bounded measurable coefficients depending
only on time. We prove that if Lu is smooth with respect to the space
variables, the same is true for u, with quantitative regularity estimates in
the scale of Sobolev spaces defined by right invariant vector fields. Moreover,
the solution and its space derivatives satisfy a 1/2-H\"older continuity
estimate with respect to time. The result is proved both for weak solutions and
for distributional solutions, in a suitable sense.
