  In the setting of a Lie group of polynomial volume growth, we derive
inequalities of Caffarelli-Kohn-Nirenberg type, where the weights involved are
powers of the Carnot-Caratheodory distance associated with a fixed system of
vector fields which satisfy the H\"ormander condition.
  The use of weak $L^p$ spaces is crucial in our proofs and we formulate these
inequalities within the framework of $L^{p,q}$ Lorentz spaces (a scale of
(quasi)-Banach spaces which extend the more classical $L^p$ Lebesgue spaces)
thereby obtaining a refinement of, for instance, Sobolev and Hardy-Sobolev
inequalities.
