Abstract. The first aim in this article is to give some sufficient conditions for a family of meromorphic mappings of a domain D in C n into P N (C) omitting hypersurfaces to be meromorphically normal. Our result is a generalization of the results of Fujimoto and Tu. The second aim is to investigate extending holomorphic mappings into the compact complex space from the viewpoint of the theory of meromorphically normal families of meromorphic mappings. §1. Introduction Classically, a family F of holomorphic functions on a domain D ⊂ C is said to be (holomorphically) normal if every sequence in F contains a subsequence which converges uniformly on all the compact subsets of D.In 1957 Lehto and Virtanen [LeVi] introduced the concept of normal meromorphic functions in connection with the study of boundary behaviour of meromorphic functions of one complex variable. Since then normal holomorphic maps has been studied intensively, resulting in an extensive development The first aim in this article is to give some sufficient conditions for a family of meromorphic mappings of a domain D in C n into P N (C) omitting hypersurfaces to be meromorphically normal or quasi-normal. These results are generalizations of the above Fujimoto's and Tu's results.The second aim of this article is to investigate extending holomorphic mappings into compact complex spaces from the viewpoint of the theory of meromorphically normal families of meromorphic mappings. In order to state our main result, we need some preliminary. First, for hypersurfaces H i (1 ≤ i ≤ q) of P N (C) with q ≥ N + 1, let Q i (1 ≤ i ≤ q) be their defining polynomials, i.e., the homogeneous polynomials without multiple factors such thatHere and below, throughout the article, we only consider homogeneous polynomials Q(z) = a ν z ν normalized so that |a ν | 2 = 1. Now we definewhere z = |z j | 2 1/2 . Next, let Λ d (S) denote the real d-dimensional Hausdorff measure of S ⊂ C n . For a formal Z-linear combination X = i∈I n i X i of analytic subsets X i ⊂ C n and for a subset E ⊂ C n , we call i∈I Λ d (X i ∩ E) (resp. i∈I n i Λ d (X i ∩ E)), the d-dimensional Lebesgue area of X ∩ E regardless of multiplicities (resp. with counting multiplicities).Now we can state our main results.

In this article we show several convergence and extension theorems for analytic hypersurfaces (not necessarily with normal crossings) and for closed pluripolar sets of complex manifolds. Moreover, a generalization of theorem of Alexander to complex spaces is given.

IntroductionThe general interpolation problem induced by a right invertible operator with a given system of initial operators possessing the c(iî)-property was introduced and investigated by D. Przeworska-Rolewicz (see [1], [2]). Moreover, Przeworska-Rolewicz also constructed some conditions for interpolation by polynomials of right invertible operators to be admissible (see In this paper, we introduce the generalized c(W)-property of right initial operators with respect to a right inverse W of a given generalized right invertible operator V. Then we give a general interpolation formula for a system of right initial operators possessing the generalized c(W)-property.

Generalizations of the theorem of Forelli to holomorphic mappings into complex spaces are given.

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