A functional calculus on the Heisenberg group and the boundary layer potential \(\square_+^{-1}\) for the \(\bar\partial\)-Neumann problem (Q1266258)

On the Heisenberg group \({\mathbb H}_n\) one has two commuting partial differential operators which are given by \(\Delta_H={1\over 2}\sum_{j=1}^{2n}X_j^2\) and \(T=-i{\partial\over\partial t}\) in the standard basis \(X_1,\dots,X_{2n},iT\) of the Lie algebra of left-invariant vector fields. The authors develop a general joint functional calculus for these operators providing formulas for the associated convolution kernels of \(\Phi(-\Delta_H,T)\) for real-valued functions \(\Phi\) on the joint spectrum. In particular, they give an explicit formula which helps solving the \(\overline\partial\)-Neumann problem on a Siegel upper-half space.