Estimates for Derivatives of the Green Functions for the Noncoercive Differential Operators on Homogeneous Manifolds of Negative Curvature

In this paper we study the Green function for a second order noncoercive differential operator L on a connected, simply connected homogeneous manifold of negative curvature. Such a manifold is a solvable Lie group S = NA, a semi-direct product of a nilpotent Lie group N and an abelian group A = R. Moreover, for an H belonging to the Lie algebra a of A, the real parts of the eigenvalues of Adexp H |n, where n is the Lie algebra of N, are all greater than 0. Conversely, every such a group equipped with a suitable left-invariant metric becomes a homogeneous Riemannian manifold with negative curvature (see [6]). On S we consider a second order left-invariant operator