Fundamental solution of a higher step Grushin type operator

We examine a class of Grushin type operators _Pk$P_k$ where k∈_N0$k\in {\mathbb{N}}_0$ defined in (1.1). The operators _Pk$P_k$ are non-elliptic and degenerate on a sub-manifold of R^N+?${\mathbb{R}}^{N+?}$. Geometrically they arise via a submersion from a sub-Laplace operator on a nilpotent Lie group of step k+1$k+1$. We explain the geometric framework and prove some analytic properties such as essential self-adjointness. The main purpose of the paper is to give an explicit expression of the fundamental solution of _Pk$P_k$. Our methods rely on an appropriate change of coordinates and involve the theory of Bessel and modified Bessel functions together with Weber's second exponential integral.