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

Abstract

By employing a new reduction procedure we derive explicit expressions for the fundamental solutions of a family Pk,λ of degenerate second order differential operators on RN+ℓ. Here λ is a complex parameter located in the strip |Re(λ)|<N+k−1. As is pointed out in [2] Pk,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 Δλ 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 Δ1 modulo the classical Szegö projection. This phenomenon had been observed before in [11].