Fundamental solution of a higher step Grushin type operator

We examine a class of Grushin type operators PkPk where k∈N0k∈N0 defined in (1.1). The operators PkPk are non-elliptic and degenerate on a sub-manifold of RN+ℓRN+ℓ. Geometrically they arise via a submersion from a sub-Laplace operator on a nilpotent Lie group of step k+1k+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 PkPk. 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.