van den Ban–Schlichtkrull–Wallach asymptotic expansions on nonsymmetric domains

Abstract

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...

Asymptotic expansions and Hua-harmonic functions on bounded homogeneous domains

“…The idea stems from the seminal papers of Schlichtkrull and van den Ban [2], [1]. The present paper is parallel to the paper of R. Penney, [22]. Some earlier incomplete versions of [22] have been known to us and we owe very much to Richard Penney for showing us the strength of the method of asymptotic expansions.…”

“…The present paper is parallel to the paper of R. Penney, [22]. Some earlier incomplete versions of [22] have been known to us and we owe very much to Richard Penney for showing us the strength of the method of asymptotic expansions. It clearly allows to go beyond symmetric domains.…”

“…Our final result is considerably stronger than Theorem 5.2 of [22] as explained below. 1 Let X = G/K be a noncompact symmetric space.…”

“…In the November 2003 issue of the Annals of Mathematics a paper by R. Penney appeared, [22]. There the idea of using asymptotic expansion of harmonic functions on bounded homogeneous domains in C n to study their boundary behavior is presented.…”

Unbounded harmonic functions on homogeneous manifolds of negative curvature

“…We include it because (a) at the time of writing, [9] was not in print, (b) our differential operator is somewhat different than that considered in [9], and (c) we require Proposition 2.8, which does not appear in [9], and whose proof requires repetition of the proof of Theorem 2.5.…”