Caffarelli-Kohn-Nirenberg inequalities on Lie groups of polynomial growth

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 Hormander condition. The use of weak L-p spaces is crucial in our proofs and we formulate these inequalities within the framework of L-p,L-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.