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

Abstract

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. K E Y W O R D SCaffarelli-Kohn-Nirenberg inequalities, interpolation, Lie group, Lorentz spaces, polynomial growth M S C ( 2 0 1 0 ) Primary: 22E30, 43A80; Secondary: 46E30, 46E35

Hardy, weighted Trudinger-Moser and Caffarelli-Kohn-Nirenberg type inequalities on Riemannian manifolds with negative curvature

“…It is a natural problem, also important for applications, to find an analogue of the above Caffarelli-Kohn-Nirenberg inequalities on Lie groups or on Riemannian manifolds. On Lie groups, we refer, for example, to [ZHD14] for Heisenberg groups, to [Yac18] for Lie groups of polynomial volume growth, to [RSY17c] and to [RS17] for stratified groups, to [RSY17a], to [RSY17b] and to [ORS17] for general homogeneous groups. On Riemannian manifolds, in [CX04] and [Mao15] the authors assuming that Caffarelli-Kohn-Nirenberg type inequalities hold, investigated the geometric property related to the volume of a geodesic ball on an n-dimensional (n ≥ 3) complete open manifold with non-negative Ricci curvature and on an n-dimensional (n ≥ 3) complete and noncompact smooth metric measure space with non-negative weighted Ricci curvature, respectively.…”