Estimates for the Poisson kernels on homogeneous manifolds of negative curvature and the boundary Harnack inequality in the noncoercive case (Q2772076)

In the paper the author considers second order left-invariant differential operators on a connected, simply connected homogeneous manifold of negative curvature. Such a manifold is a solvable Lie group \(S=NA\), being a semidirect product of a nilpotent Lie group \(N\) and an abelian group \(A=\mathbb{R}^+\). Using a probabilistic technique the author obtains upper and lower estimates for the Poisson kernels of some classes of left-invariant differential operators. For the noncoercive operator the boundary Harnack inequality is proved which turns out to be the same as in the coercive case. The analogous boundary Harnack inequality for coercive operators has been proved by \textit{E. Damek} [Colloq. Math. 72(2), 229-249 (1997; Zbl 0878.22007)].