Soit X un groupe abélien localement compact séparé et dénombrable á l’infini. On désignera par ξ la mesure de Haar sur X. Un noyau de convolution réel N (c’est-à-dire, une mesure de Radon réelle) sur X sera dit de type logarithmique s’il existe un semi-groupe (αt)t≥0 vaguement continu, markovien et récurrent des noyaux de convolution (non-négatifs) tel que limt → ∞ (vaguement) et, pour toute (pour la notation, voir § 2),

1. Let Ω be a bounded domain of Rn with boundary ∂Ω and let E be a compact subset of Ω. We assume that both ∂Ω and E have positive capacity. The norm and the inner product in L2(Ω) are simply denoted by ‖ ‖ and (,) respectively.

Urbanik introduces in [16] and [17] the classes Lm and L∞ of distributions on R1 and finds relations with stable distributions. Kumar-Schreiber [6] and Thu [14] extend some of the results to distributions on Banach spaces. Sato [7] gives alternative definitions of the classes Lm and L∞ and studies their properties on Rd. Earlier Sharpe [12] began investigation of operator-stable distributions and, subsequently, Urbanik [15] considered the operator version of the class L on Rd. Jurek [3] generalizes some of Sato’s results [7] to the classes associated with one-parameter groups of linear operators in Banach spaces. Analogues of Urbanik’s classes Lm (or L∞) in the operator case are called multiply (or completely) operator-selfdecomposable. They are studied in relation with processes of Ornstein-Uhlenbeck type or with stochastic integrals based on processes with homogeneous independent increments (Wolfe [18], [19], Jurek-Vervaat [5], Jurek [2], [4], and Sato-Yamazato [9], [10]). The purpose of the present paper is to continue the preceding papers, to give explicit characterizations of completely operator-selfdecomposable distributions and operator-stable distributions on Rd, and to establish relations between the two classes. For this purpose we explore the connection of the structures of these classes with the Jordan decomposition of a basic operator Q.

This paper is a continuation of our previous memoir [28], hereafter referred to as I, and constitutes the second chapter of this series. As stated in I, our aim in this series is to examine properties of a polynomial sequence with several variables satisfying an addition formula by means of the down-ladder, and to give a generalization of so called classical polynomials.

Dans tout ce qui suit, les varietés qui interviendront seront définies sur le corps des nombres complexes C. L’objet de cet article est de préciser le résultat suivant obtenu dans [B-2] et [K-1]:

Théorème 0. Soit X une variété protective normale de dimension 3, dont les singularités sont canoniques au sens de [R-1]. Soient e le p.p.c.m. des indices des points singuliers de X. Soit R un diviseur de Cartier sur X, tel que le bidual de soit isomorphe à Supposons que R soit numériquement positif et que R3 > 0. Alors il existe un entier n0 > 0 tel que pour tout le système linéaire |nR| soit sans point base; en particulier l’anneau est de type fini sur C.

In the present note, we prove that every simple CM-type is non-degenerate (i.e. the rank is maximal) if the dimension of corresponding abelian varieties is a prime. This follows directly from the argument of Tankeev [5], in which he has treated the 5-dimensional case.

Recently, S. G. Tankeev and K. A. Ribet have established similar results for more general types of abelian varieties (see [3], [4], [6]).

The study of finiteness for Riemannian manifolds, which has been done originally by J. Cheeger [5] and A. Weinstein [13], is to investigate what bounds on the sizes of geometrical quantities imply finiteness of topological types, —e.g. homotopy types, homeomorphism or diffeomorphism classes-— of manifolds admitting metrics which satisfy the bounds. For a Riemannian manifold M we denote by RM and KM respectively the curvature tensor and the sectional curvature, by Vol (M) the volume, and by diam(M) the diameter.