Degenerate $p$-Laplacian Operators and Hardy Type Inequalities on H-Type Groups

Journal article, 2010

Let G  be a step-two nilpotent group of H-type with Lie algebra G=V⊕t  . We define a class of vector fields X={X j }  on G  depending on a real parameter k≥1  , and we consider the corresponding p  -Laplacian operator L p,k u=div X (|∇ X u| p−2 ∇ X u)  . For k=1  the vector fields X={X j }  are the left invariant vector fields corresponding to an orthonormal basis of V  ; for G  being the Heisenberg group the vector fields are the Greiner fields. In this paper we obtain the fundamental solution for the operator L p,k   and as an application, we get a Hardy type inequality associated with X  .