Abstract. In this paper we study the structure of closed weakly dense ideals in Privalov spaces N p (1 < p < ∞) of holomorphic functions on the disk D : |z| < 1. The space N p with the topology given by Stoll's metric [21] becomes an F -algebra. N. Mochizuki [16] proved that a closed ideal in N p is a principal ideal generated by an inner function. Consequently, a closed subspace E of N p is invariant under multiplication by z if and only if it has the form IN p for some inner function I. We prove that if M is a closed ideal in N p that is dense in the weak topology of N p , then M is generated by a singular inner function. On the other hand, if Sµ is a singular inner function whose associated singular measure µ has the modulus of continuity O(t (p−1)/p ), then we prove that the ideal SµN p is weakly dense in N p . Consequently, for such singular inner function Sµ, the quotient space N p /SµN p is an F -space with trivial dual, and hence N p does not have the separation property.

Abstract. For a meromorphic function /(z) defined in the unit disc D : lzl < 1 on the complex z-plane, z = x * iy, we denote its spherical derivative bv f#(r) and introduce the differentiable form dpyQ) = (1 -lzl2)lf+(z)l2dxdy. We prove that f(z) has the uniformly bounded characteristic if and only if the measur" WQ) is the Carleson measure. This result answers a question posed by S. Yamashita in Internat.

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We establish the theorems that give necessary and sufficient conditions for an arbitrary function defined in the unit disk of complex plane in order to has boundary values along classes of equivalencies of simple curves. Our results generalize the well-known theorems on asymptotic and angular boundary behavior of meromorphic functions (Lindölf, Lehto-Virtanen, and Seidel-Walsh type theorems). The results are applied to the study of boundary behavior of meromorphic functions along curves using P −sequences, as well as in the proof of the uniqueness theorem similar toŠaginjan's one. Constructed examples of functions show that the results cannot be improved.1 2010 Mathematics Subject Classification: Primary 30D40; Secondary 51K99 One of the classical results of The theory of cluster sets related to the asymptotic behavior is the theorem of Lindelöf on angular boundary values of analytic functions [28] (or, see [54]). Further interesting results on the boundary behavior of analytic functions along simple curves were obtained by Seidel [46] and Seidel and Walsh [47] (see also [29]). Lehto and Virtanen's result from [27], which is a transfer of the results of Lindelöf and Seidel and Walsh to the class of normal meromorphic functions in the unit disc, the class usually denoted by N , prompted a further intensive research in the Theory of cluster sets. These investigations were also related to the boundary behavior of functions along sequences of points on the one hand, and on the boundary behavior of harmonic, subharmonic, continuous functions, and normal quasiconformal and equimorphic mappings along (non-)tangential simple curves (see References).While most of these papers concern the boundary properties of functions along simple curves which are at the finite Fréshet distance or finite Hausdorff distance (see eg.[8]), in this paper we define a relation of equivalence in the family of all simple curves in the unit disc which terminate in the same point on the boundary, and study the boundary behavior of functions along classes of equivalence. We also offer an example of two simple curves ending in a point of the boundary of the unit disc which belong to the same equivalence class, such that their Fréshet distance is infinite. This the content of Lemma 3.3. Thus, our results in the paper are generalization of some known results. Namely, using the mentioned relation of equivalence we prove the theorem that give necessary and sufficient conditions for an arbitrary function defined in the unit disk to has a curvilinear boundary value (see Theorem 4.1). This theorem is used in proof of Theorem 4.2, which shows that for an arbitrary function in the unit disk holds an analogue of Theorem 1 in [27] concerning the meromorphic functions. As follows from our Theorem 4.1, the normality along simple curves is a necessary condition for the existence of curvilinear boundary value of functions. In Section 5 we study the normality and boundary behavior of meromorphic functions using the P −sequences. We emphasis that the P −sequences p...