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 spaces is crucial in our proofs and we formulate these inequalities within the framework of Lorentz spaces (a scale of (quasi)‐Banach spaces which extend the more classical Lebesgue spaces) thereby obtaining a refinement of, for instance, Sobolev and Hardy–Sobolev inequalities.