Explicit fundamental solution for the operator $L+α|T|$ on the Gelfand pair $(\mathbb{H}_{n},U(n))$
By means of the spherical functions associated to the Gelfand pair $(\mathbb{H}_{n},U(n))$ we define the operator $L+α|T|$, 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 $α<n$ we use the Integral Representation Theorem to obtain a more detailed expression. Finally, we remark that when $α=0$ we recover the fundamental solution for the Heisenberg sublaplacian given by Folland.
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