The inverse of a parameter family of degenerate operators and applications to the Kohn-Laplacian.

By employing a new reduction procedure we derive explicit expressions for the fundamental solutions of a family P k , λ of degenerate second order differential operators on R N + ℓ . Here λ is a complex parameter located in the strip | Re ( λ ) | < N + k − 1 . As is pointed out in [2] P k , 0 has a geometric background and arises as a Grushin-type operator induced by a sub-Riemannian structure on a k + 1 -step nilpotent Lie group. Our method leads to new formulas for the inverse of the Kohn-Laplacian Δ λ which has been widely studied before in the framework of pseudo-convex domains and CR geometry. As an application we show that in all cases the fundamental solutions have a meromorphic extension in the parameter λ to C ∖ Q . All poles are simple and Q ⊂ R is an explicitly given discrete set. We recover the invertibility of Δ 1 modulo the classical Szegö projection. This phenomenon had been observed before in [11] . [ABSTRACT FROM AUTHOR]