Degenerate p-Laplacian operators on H-type groups and applications to Hardy type inequalities

Abstract

Let $\mathbb{G}$ be a step-two nilpotent group of H-type with Lie algebra $\mathfrak{G}=V\oplus \mathfrak{t}$. We define a class of vector fields $X=\{X_j\}$ on $\mathbb{G}$ depending on a real parameter $k\ge 1$, and we consider the corresponding $p$-Laplacian operator $L_{p,k} u= \text{div}_X (|\na_{X} u|^{p-2} \na_X u)$. For $k=1$ the vector fields $X=\{X_j\}$ are the left invariant vector fields corresponding to an orthonormal basis of $V$, for $k=2$ and $\mathbb{G}$ being the Heisenberg group they are introduced by Greiner \cite{Greiner-cjm79}. 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 $X$.

Canadian Math. J., to appear