COLLOQUIUM MATHEMATICUM VOL. 118 2010 NO. 1 ESTIMATES FOR THE POISSON KERNEL ON HIGHER RANK NA GROUPS BY RICHARD PENNEY (West Lafayette, IN) and ROMAN URBAN (Wroclaw) Abstract. We obtain an estimate for the Poisson kernel for the class of second order left-invariant differential operators on higher rank NA groups. The authors would like to dedicate this paper to the memory of Andrzej Hulanicki. As is clear from the bibliography, this work owes much to the influence of him and his co-workers. Indeed, this whole area of exploration was initiated by this group. The current work could not have been done without the foundation they laid. However, our debt goes far beyond this. The second author was a stu- dent of Andrzej’s student, Ewa Damek. The first author came to Poland for the first time in 1976 at Andrzej’s invitation. Since then he has visited Poland regularly, at first to attend conferences, and later to do mathemat- ics both with Andrzej and others. This collaboration has been one of the most rewarding experiences of his career. In the process Andrzej and his wife Barbara became good friends of his. He spent many memorable hours with them, both in Poland and elsewhere, sharing a good meal (cooked by Barbara) and discussing mathematics, life, etc. over a glass of good wine or vodka. Andrzej will be dearly missed. 1. Statement of the result. Let S be a semidirect product S = N ⋊ A where N is a connected and simply connected nilpotent Lie group and A is isomorphic with R k . For g ∈ S we let n(g)= n and a(g)= a denote the components of g in this product so that g =(n,a). We assume that there is a basis X 1 ,...,X m for n that diagonalizes the A-action. Let ξ 1 ,...,ξ m ∈ a ∗ = R k be the corresponding roots, i.e., for every H ∈ a, [H,X j ]= ξ j (H )X j ,j =1,...,m. As in [3], we assume that there is an element H ∈ R k such that ξ j (H ) > 0 for 1 ≤ j ≤ m. 2010 Mathematics Subject Classification : 43A85, 31B05, 22E25, 22E30, 60J25, 60J60. Key words and phrases : Poisson kernel, harmonic functions, solvable Lie groups, left in- variant operators, homogeneous group, Brownian motion, higher rank NA groups. DOI: 10.4064/cm118-1-14 [259] c  Instytut Matematyczny PAN, 2010