An upper bound for the Poisson kernel on higher rank \(NA\) groups (Q645040)

Let \(S\) be a solvable higher rank \(NA\) group, i.e., \(S\) is a semidirect product \(S=N\rtimes A\) where \(N\) is a connected and simply connected nilpotent Lie group and \(A\) is isomorphic with \({\mathbb R}^k\), \(k\geq 2\). For \(g\in S\) let \(n(g)=n\) and \(a(g)=a\) denote the components of \(g\) in this product so that \(g=(n,a)\). Let \(\mathfrak n\) and \(\mathfrak a\) be the Lie algebras of \(N\) and \(A\) respectively. We assume that there is a basis \(X_1, \dots, X_m\) for \(\mathfrak n\) that diagonalizes the \(A\)-action. Let \(\xi_1, \dots, \xi_n\in{\mathfrak a}^*=\mathbb R^k\) be the corresponding roots, i.e., for every \(H\in {\mathfrak a}\), \([H, X_j]= \xi_j(H) X_j\), \(j=1, \dots, m\). Let, for \(\alpha=(\alpha_1, \dots, \alpha_n)\in \mathbb R^k\) and real \(d_j\)'s, \[ {\mathcal L}_\alpha:=\sum_{j=1}^r \left( e^{2\xi_j(a)} X_j^2+ d_j e^{\xi_j(a)} X_j\right) +\Delta_\alpha, \] where \[ \Delta_\alpha=\sum_{i=1}^k \left(\partial_{a_i}^2 -2\alpha_i \partial_{a_i}\right), \] and \(X_1, \dots, X_r\) satisfy the Hörmander condition, i.e., they generate the Lie algebra \(\mathfrak n\). The main result of the paper under review is an estimate for the Poisson kernel of the differential operators \({\mathcal L}_\alpha\). This estimate is much better than the estimate of the authors' paper [Colloq. Math. 118, No.~1, 259--281 (2010; Zbl 1190.43007)].