Zero sets of holomorphic functions in the unit ball with slow growth

We study the zero-varieties of holomorphic functions in the unit ball satisfying the growth condition log |f(z)|≤c _fλ(|z|), where λ:(0,1)→ℝ⁺ is a positive increasing function. We obtain some sufficient conditions on an analytic variety to be defined by such a function. Some results for the particular case λ(r)=log(e/(1−r)), corresponding to the classA ^−∞, generalize those of B. Korenblum in one variable. Both authors supported by DGICYT grant PB92-0804-C02-02.     

Universality of holomorphic functions bounded on closed sets

In this note, the existence of translation-universal entire functions which are bounded on certain closed subsets is characterized in terms of topological and geometrical properties of such subsets. Corresponding results are also stated in the space of holomorphic functions on the unit disk and in the space of harmonic functions on the plane. Moreover, it is shown the existence of entire functions which are bounded on many rays and, simultaneously, are universal with respect to a prescribed infinite-order differential operator.     

Envelopes of open sets and extending holomorphic functions on dual Banach spaces

We investigate certain envelopes of open sets in dual Banach spaces which are related to extending holomorphic functions. We give a variety of examples of absolutely convex sets showing that the extension is in many cases not possible. We also establish connections to the study of iterated weak^∗ sequential closures of convex sets in the dual of separable spaces.     

Fréchet-valued holomorphic functions on compact sets in (DFN)-spaces

We establish the equivalence between the weak holomorphicity and holomorphicity of Fréchet-valued functions on compact polydisks in (DFN)-spaces. Moreover, the relations between separately holomorphic functions and holomorphic functions on compact polydisks in (DFN)-spaces are also given. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 11, pp. 1578–1584, November, 2008.     

Leafwise holomorphic functions

It is a well-known and elementary fact that a holomorphic function on a compact complex manifold is necessarily constant. The purpose of the present article is to investigate whether, or to what extent, a similar property holds in the setting of holomorphically foliated spaces.