The Poisson Integral for Generalized Half-Planes and Bounded Symmetric Domains

Abstract

In the case of the upper half-plane, or the unit disc in the theory of one complex variable, there is a simple classical way of constructing the Poisson kernel from the Szegb kernel function. This construction was successfully used in the more general cases of the four classical types of bounded symmetric domains by L. K. Hua [9], and in the case of wedge domains (i.e., tube domains over convex cones) by E. M. Stein, G. Weiss, and M. Weiss [18], and the author [11]. Recently S. G. Gindikin [4] determined the Szegb kernel function for a wider class of generalizations of the classical upper half-plane than wedge domains, the so-called Siegel domains of type ii introduced by I. I. PjatetskiiSapiro [16]. The purpose of the present paper is to study the Poisson kernel obtained by the classical construction for these domains. First, in ? 2, we consider the case of a general Siegel domain of type II. The main result here is Theorem 2.4 which states that the Poisson kernel is an approximate identity. This theorem contains as a special case the analogous theorem of [18], while its proof given here is considerably simpler than the special proofs in [18] and [11]. As applications of Theorem 2.4, we establish a few simple facts about HI-spaces on D. We do not pursue this investigation very far; the present paper is rather meant to lay the ground work for a future study of finer questions of analysis on generalized half-planes. In ? 3, we consider the case of a Siegel domain of type II which is symmetric in the sense of E. Cartan. It is known [16], [12] that every bounded symmetric domain has a realization of this kind. We prove that in this case the Poisson kernel is harmonic in the sense of the theory of symmetric spaces. This identifies our Poisson kernel with a kernel studied in more general contexts by R. Hermann [7] and H. Furstenberg [3]. In the case of symmetric domains, our results go farther than those in [3] and [7] in two directions. We obtain fairly strong results on the boundary behaviour of the Poisson integral (?? 2 and 4) and we find easily computable explicit formulas for the Poisson kernel (?5). In ? 4, we consider bounded symmetric domains in the canonical Harish-