A Functional Calculus on the Heisenberg Group and the Boundary Layer Potential □−1+for the ∂-Neumann Problem

There are two natural commuting self-adjoint operators in the enveloping algebra of the Heisenberg group: the Heisenberg sublaplacianΔHand the central elementT=−i∂/∂t. The joint spectral theory of these operators is investigated by means of the Laguerre calculus. Explicit convolution kernels are obtained for a large class of functionsΦ(−ΔH, T). In particular we find the kernels of the operators□+, α=−ΔH−αT+T2−Tthat occur in the Kohn solution of the ∂-Neumann problem for the associated Siegel domain.