Explicit fundamental solution for the operator $L+\alpha |T|$ on the Gelfand pair $({\mathbb{H}}_n,U(n))$

Abstract

By means of the spherical functions associated to the Gelfand pair $({\mathbb{H}}_n,U(n))$ we define the operator $L+\alpha |T|$, where $L$ denotes the Heisenberg sublaplacian and $T$ denotes de central element of the Heisenberg Lie algebra, we establish a notion of fundamental solution and explicitly compute in terms of the Gauss hypergeometric function. For $\alpha <n$ we use the Integral Representation Theorem to obtain a more detailed expression. Finally, we remark that when $\alpha =0$ we recover the fundamental solution for the Heisenberg sublaplacian given by Folland.

19 pages