Estimates for the mixed derivatives of the Green functions on homogeneous manifolds of negative curvature

We consider the Green functions for second-order left-invariant differential operators on homogeneous manifolds of negative curvature, being a semi-direct product of a nilpotent Lie group $N$ and $A=mathbb{R}^+$. We obtain estimates for mixed derivatives of the Green functions both in the coercive and non-coercive case. The current paper completes the previous results obtained by the author in a series of papers [14,15,16,19].