  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 $X$.

Analysis of PDEs;Classical Analysis and ODEs