Fredholm property of a family of operators on the five-dimensional Heisenberg group. (Q1418009)

Let \({\mathbb H}\;(\approx {\mathbb C}^2 \times{\mathbb R})\) be the 5-dimensional Heisenberg group. The Heisenberg Laplacian arises naturally in the study of the \({\mathbb C}{\mathbb R}\)-structure of \({\mathbb H}\). In the special case of the Dirichlet problem for the Laplacian, a spherical harmonic expansion reduces the initial 5-dimensional problem to an infinite number of 2-dimensional Dirichlet equations which are indexed by the bi-degree \((p,q)\) of spherical harmonics. Fundamental solutions are obtained for each of the partial differential operators in these problems, thus reducing them further via layer potentials to one-dimensional integral operators \(K^{(p,q)}\) on a half-circle \(C^+\). The operators \(K^{(p,q)}\) are bounded from some weighted \(L^2\)-space \(L\) to an appropriate weighted Sobolev space \(W\). The main result of this paper states that the integral operators \(K^{(p,q)}\) are Fredholm. Fredholm property is shown by constructing a regularizer \(R^{(p,q)}:H\to L\). The operators \(R^{(p,q)}\) are obtained by inverting \(K^{(p,q)}\) separately in the interior and near the end-points of \(C^+\).