Noncoercive differential operators on homogeneous manifolds of negative curvature and their Green functions (Q2717708)

Let \(S\) be a connected, simply connected homogeneous manifold of negative curvature. Such a manifold is a solvable Lie group \(S=NA\), a semidirect product of a nilpotent Lie group \(N\) and an abelian group \(A={\mathbb R}^+\). Let \({\mathcal L}=Y_0^2+Y_1^2+\dots+Y_m^2+Y\) be left-invariant second order differential operators, where \(Y, Y_0, \dots, Y_m\) are in the Lie algebra \(\mathcal S\) of \(S\) and \(Y_0, Y_1, \dots, Y_m\) generate the Lie algebra \(\mathcal S\). Let \(\pi: S\to A=S/N\) be the canonical homomorphism. Then the image of \(\mathcal L\) under \(\pi\) is a second order differential operator on \({\mathbb R}^+\): \((a \partial_a)^2-\gamma a \partial_a\) where \(\gamma\in{\mathbb R}\). The operator \({\mathcal L}={\mathcal L}_\gamma\) is noncoercive (there is no \(\varepsilon>0\) such that \({\mathcal L}+\varepsilon I\) admits the Green function) if and only if \(\gamma=0\). Let \({\mathcal G}(xa,yb)\) be the Green function for \(\mathcal L\) and let \({\mathcal G}(x,a)={\mathcal G}(xa,e)\), where \(x\in N\), \(a\in A\), \(e\) is the identity element of the group \(S\). In the article [Colloq. Math. 73, 229-249 (1997; Zbl 0878.22007)] \textit{E.~Damek} proved upper and lower estimates for the function \({\mathcal G}(x,a)\) for the operator \({\mathcal L}_\gamma\) with \(\gamma>0\). In the paper under review the author proves similar estimates for the noncoercive operator \({\mathcal L}_\gamma\) with \(\gamma=0\). These estimates cannot be obtained from Damek's estimate by taking the limit and so require new methods. In this paper the author makes use of a probabilistic method introduced in [\textit{E.~Damek} and \textit{A.~Hulanicki}, Stud. Math. 101, 33-68 (1991; Zbl 0811.43001)].