Estimates for the Poisson kernel and the evolution kernel on nilpotent meta-abelian groups

Let $S$ be a semi direct product $S=N\rtimes A$ where $N$ is a connected and simply connected, non-abelian, nilpotent meta-abelian Lie group and $A$ is isomorphic with $\R^k,$ $k>1.$ We consider a class of second order left-invariant differential operators on $S$ of the form $\mathcal L_\alpha=L^a+\Delta_\alpha,$ where $\alpha\in\R^k,$ and for each $a\in\R^k,$ $L^a$ is left-invariant second order differential operator on $N$ and $\Delta_\alpha=\Delta-<\alpha,\nabla>,$ where $\Delta$ is the usual Laplacian on $\R^k.$ Using some probabilistic techniques (e.g., skew-product formulas for diffusions on $S$ and $N$ respectively) we obtain an upper bound for the Poisson kernel for $\mathcal L_\alpha.$ We also give an upper estimate for the transition probabilities of the evolution on $N$ generated by $L^{\sigma(t)},$ where $\sigma$ is a continuous function from $[0,\infty)$ to $\R^k.$