In this paper we consider the modified wave equation associated with a class of radial Laplacians L generalizing the radial part of the Laplace-Beltrami operator on hyperbolic spaces or Damek-Ricci spaces. We show that the Huygens' principle and the equipartition of energy hold if the inverse of the Harish-Chandra c-function is a polynomial and that these two properties hold asymptotically otherwise. Similar results were established previously by Branson, Olafsson and Schlichtkrull in the case of noncompact symmetric spaces.

In this note, Kelvin transform is introduced in the framework of Dunkl-Clifford analysis. It is shown that this transform preserves the class of Dunkl monogenic functions. As an application, we use it to generate Dunkl monogenic polynomials by a classical process due to Maxwell.

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