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