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The results of [ 1 ] concerning tight C0* limit of the Poisson kernel of a tube domain, as its parameter converges to a point on the cone boundary, are extended under certain hypotheses to Siegel domains of type II. In the case where the domain is polytopic, almost everywhere convergence of the Lp Poisson integral to its boundary values is obtained. Examples and further conjectures conclude the paper. I. Notation and basics. In this paper we extend some of the results of [1] to Siegel domains of type II (which we will henceforth call "Siegel domains"). This paper is a sequel to [1], from which we draw our basic notation. Also we wiU refer freely to paraUel results and proofs in that paper. We wiU make use of two notational conveniences. The lower-case letter c, with or without subscripts, wül always denote a positive constant or function dependent only on the dimensions m and « and the original Siegel domain Í2 = J2r .j, under consideration (see below). Secondly, we wiU omit explicit references to degenerate cases (such as « = 0) while including them in our theorems: the interpretation of the 0 x k matrices, 0x0 determinants, etc., arising in these cases is obvious. 1.1. Definitions. Let Cm = Rm © /Rm and C" be finite-dimensional vector spaces over C, with (fixed) inner products induced by the direct sum, and Rm a (fixed) real form of Cm. A Siegel domain of class (m, ri) is a domain £2 of Cm+n = Rm © iRm © C" defined as foUows [2], [4] : (1) Í2 = nr<¡, ={(x + iy + i*(z), z): jc G Rm,y G T, z G C}, where *: C" x C" —»• Cm is a Hermitian form (relative to Rm) satisfying, for the proper cone T, Received by the editors October 22, 1974. AMS (MOS) subject classifications (1970). Primary 31B10, 52A20; Secondary 31B2S, 32F05.

Limit properties of Poisson kernels of Siegel domains of type II