Integral representation formulas for holomorphic functions on analytic subvarieties of domains of C" are derived. These formulas generalize the Cauchy-Fantappie formula and the Weil formula for analytic polyhedra. The kernels we obtain are explicitly defined.Introduction. In recent years integral formulas and their applications have attracted a lot of attention in several complex variables; see for example [4,5,7,8,9, 10] and the most relevant to our work papers of Stout [15], Palm [11] and Henkin and Leiterer [6].In this paper we develop analogues of Cauchy-Fantappie Kernels for analytic subvarieties of domains of C". First let us recall the CauchyFantappie formula. Let ΰcC" be a bounded domain with smooth boundary and let γ: (dD) X D -> C" be a smooth function so that Some of the interesting features of Theorem I.I is on the one hand the explicit form of the kernel K^ξ, z) and on the other hand the fact that M is allowed to have, finitely many, singular points.The case m = 0 of Theorem LI is the formula (2) and the case m = 1 of it was obtained by Stout [15]. In fact Stout's paper was the starting point of this work. The kernel obtained by Stout coincides with ours in the case m = 1; this is not immediate however and in §11 we show that they are indeed the same.The proof of Theorem I.I (given in this paper) is an extension of Stout's proof from the case m = 1 to the general case. A second proof of Theorem I.I is contained in Hatziafratis [3].In §111 we develop a Weil type integral formula for analytic polyhedra on analytic varieties. The main result of this section is Theorem III.l which generalizes the Weil integral formula for analytic polyhedra in C" (see [14, 16]). To obtain this result we combine results from §1 together with some standard techniques contained for example in Range-Siu [12].As we pointed out before the results in this paper are related to those of Palm [11] and Henkin and Leiterer [6]. The setting of Henkin and Leiterer [6] is more general than ours (we allow, however, finitely many singular points on M). On the other hand our results are more explicit than theirs. In fact we do not know the relation between our results and those of Henkin-Leiterer and Palm.Acknowledgment. I would like to express my warmest thanks to Professor Alexander Nagel for very helpful discussions.

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We consider Bergman spaces and variations of them on domains  in one or several complex variables. For certain domains  we show that the generic function in these spaces is totaly unbounded in  and hence non-extendable. We also show that generically these functions do not belong -not even locally -in Bergman spaces of higher order. Finally, in certain domains  , we give examples of bounded non-extendable holomorphic functions -a generic result in the spaces ) (  s of holomorphic functions in  whose derivatives of order s  extend continuously to  (    s 0 ). AMS classification numbers:Primary 30H20, 32A36, Secondary 32D05, 32T05, 54E52.

We compare the geometric concept of strict convexity of open subsets of K" with the analytic concept of 2-strict convexity, which is based on the defining functions of the set, and we do this by introducing the class of 2A'-strictly convex sets. We also describe an exhaustion process of convex sets by a sequence of 2-strictly convex sets.2000 Mathematics subject classification: primary 52A20.