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

TL;DR

This paper derives an explicit fundamental solution for the operator $L+ extstylerac{eta}{|T|}$ on the Gelfand pair $( ext{Heisenberg group}, U(n))$, generalizing Folland's solution for the sublaplacian.

Contribution

It introduces a new explicit fundamental solution for the operator $L+rac{eta}{|T|}$ on the Gelfand pair, extending known results for the Heisenberg sublaplacian.

Findings

Explicit fundamental solution expressed via Gauss hypergeometric function.

Detailed expression obtained for $eta<n$ using the Integral Representation Theorem.

Recovers Folland's fundamental solution when $eta=0$.

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.