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IntroductionLet X = G/K be a homogeneous Riemannian manifold where G is the identity component of its isometry group. A C ∞ function F on X is harmonic if it is annihilated by every element of D G (X), the algebra of all G-invariant differential operators without constant term. One of the most beautiful results in the harmonic analysis of symmetric spaces is the Helgason conjecture, which states that on a Riemannian symmetric space of noncompact type, a function is harmonic if and only if it is the Poisson integral of a hyperfunction over the Furstenberg boundary G/P o where P o is a minimal parabolic subgroup. (See [14], [17].) One of the more remarkable aspects of this theorem is its generality; one obtains a complete description of all solutions to the system of invariant differential operators on X without imposing any boundary or growth conditions.If X is a Hermitian symmetric space, then one is typically interested in complex function theory, in which case one is interested in functions whose boundary values are supported on the Shilov boundary rather than the Furstenberg boundary. (The Shilov boundary is G/P where P is a certain maximal parabolic containing P o .) In this case, it turns out that the algebra of G invariant differential operators is not necessarily the most appropriate one for defining harmonicity. Johnson and Korányi [16], generalizing earlier work of Hua [15], , and , introduced an invariant system of second order differential operators (the HJK system) defined on any Hermitian symmetric space. In [9], we noted that this system could be defined entirely in terms of the geometric structure of X as Thus, in the tube case, these results yield a complete description of all solutions to the Hua system, while in the nontube case, we lack only a characterization of those hyperfunctions on the Shilov boundary whose Poisson integrals are Hua-harmonic.Since the Hua system is meaningful for any Kähler manifold X, it seems natural to ask to what extent these results are valid outside of the symmetric case. One might, for example, consider homogeneous Kähler manifolds. There is a structure theory for such manifolds that was proved in special cases by Gindikin and Vinberg [13] and in general by Dorfmeister and Nakajima [10] that states that every such manifold admits a holomorphic fibration whose base is a bounded homogeneous domain in C n , and whose fiber is the product of a flat, homogeneous Kähler manifold and a compact, simply connected, homogeneous, Kähler manifold. It follows that one should first consider generalizations to the class of bounded homogeneous domains in C n .This problem was considered in [9] and [25]. In both of these works, however, extremely restrictive growth conditions were imposed on the solutions: in [9] the solutions were required to be bounded and in [25] an H 2 type condition was imposed.The technical difficulties involved in eliminating these growth assumptions at first seem daunting. In the nonsymmetric case, K can be quite small. Thus, arguments which are based on con...

We obtain an estimate for the Poisson kernel for the class of second order left-invariant differential operators on higher rank N A 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 student 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 mathematics 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.