Euclidean spherical harmonics and the Heisenberg Laplacian: A new family of kernals

This work investigates the use of Euclidean spherical harmonics in the context of boundary value problems for the Laplacian $L$ on the Heisenberg group of dimension five. We express the -Neumann boundary conditions on the Heisenberg ball as an infinite number of two-dimensional systems.; We then invert the Folland-Stein fundamental solution to obtain a family of kernels Kp,q indexed by the bi-degree of spherical harmonics. These kernels are fundamental solutions for the operators Lp,q obtained by restricting $L$ on an appropriate space of functions.; Finally, we establish the Fredholm property on a weighted L 2 space of the first layer potentials induced by these kernels. This is done by “regularizing” Kp,q, that is, obtaining an operator Rp,q so that R^p,qK^p,q=I+compact andK ^p,qR^p,q =I+compact. Finding the Rp,qs amounts to the inversion of symbols arising from Mellin operators.; In addition, various other mapping properties of the kernels Kp,q are investigated. We also deduce a formula for the Green's function for spherical functions on the Heisenberg ball.