Explicit fundamental solution for the operator \n \n \n \n $$L+\\alpha |T|$$\n \n \n L\n +\n α\n |\n T\n |\n \n

Abstract

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