Representation of holomorphic functions on coverings of pseudoconvex domains in Stein manifolds via integral formulas on these domains

The classical integral representation formulas for holomorphic functions defined on pseudoconvex domains in Stein manifolds play an important role in the constructive theory of functions of several complex variables. In this paper, we will show how to construct similar formulas for certain classes of holomorphic functions defined on coverings of such domains.     

Proper holomorphic maps between Reinhardt domains and strictly pseudoconvex domains

LetR be a Reinhardt domain andD a bounded simply connected strictly pseudoconvex domain withC ^∞ boundary. We prove that any proper holomorphic mapF:R →D is, up to biholomorphism ofD, of the form \((z_1^{d_1 } , z_2^{d_2 } , \ldots , z_n^{d_n } )\) withd ₁,d ₂,…,d _n ∈ IN.     

On proper holomorphic mappings of strictly pseudoconvex domains

Siberian Mathematical Journal -     

Boundary jets of holomorphic maps between strongly pseudoconvex domains

In this paper we consider jets taken at a fixed boundary point of germs of holomorphic diffeomorphisms which send one strongly pseudoconvex domain into another. We completely describe possible first and second jets and conditions of extremality in terms of the Chern-Moser normal forms of the domains.     

Approximation of holomorphic functions by Taylor-Abel-Poisson means

We investigate the problem of approximation of functions ƒ holomorphic in the unit disk by means A _{ρ, r} (f) as ρ → 1−. In terms of the error of approximation by these means, a constructive characteristic of classes of holomorphic functions H _p ^r Lipα is given. The problem of the saturation of A _{ρ, r} (f) in the Hardy space H _p is solved. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 9, pp. 1253–1260, September, 2007.     

Approximation and interpolation on complex algebraic varieties in pseudoconvex domains

This paper investigates the twin problems of approximation and interpolation employing weighted integral representation formulas of Berndtsson-Andersson. These interpolation techniques are applied to extend local rational approximation estimates from complex algebraic complete intersection varietyX, of pure dimensionn, into a strictly pseudoconvex semi-local domainD in the ambient space ℂ ^N withN=n+p, p>0. We also use weighted intergral representation formulas to provide criteria for both Montessus-type convergence and convergence in logarithmic capacity of diagnonal rational sequences. The logarithmic capacity we use is carefully defined via Siciak’sL-family of extremal plurisubharmonic functions. This author is grateful to the NSA for partial support during the period of this research.     

Spaces of holomorphic functions on non-balanced domains

This paper studies the coincidence of the _τω${\tau }_{\omega }$ and _τδ${\tau }_{\delta }$ topologies on the space of holomorphic functions defined on an open subset U   of a Banach space. Dineen and Mujica proved that _τω=_τδ${\tau }_{\omega }={\tau }_{\delta }$ when U   is a balanced open subset of a separable Banach space with the bounded approximation property. Here, we study the _τω=_τδ${\tau }_{\omega }={\tau }_{\delta }$ problem for several types of non-balanced domains U.     

On holomorphic functions of slow growth on coverings of strongly pseudoconvex manifolds

In this paper we extend some results of the paper [M. Gromov, G. Henkin, M. Shubin, Holomorphic L²-functions on coverings of pseudoconvex manifolds, Geom. Funct. Anal. 8 (1998) 552-585].     

Gradients of inner functions in strictly pseudoconvex domains

We investigate the unbalanced ordinary partition relations of the form λ → (λ, α)² for various values of the cardinal λ and the ordinal α. For example, we show that for every infinite cardinal κ, the existence of a κ+-Suslin tree implies κ⁺ ? (κ⁺, log _κ (κ⁺) + 2)². The consistency of the positive partition relation b → (b, α)2 for all α < ω₁ for the bounding number b is also established from large cardinals.     

Hilbert spaces of holomorphic functions on bounded domains

In [6] the notion of Hilbert-Stein complex spaces was introduced. The purpose of this note is to illustrate the conjecture that any Stein space whose points can be separated by bounded holomorphic functions, is Hilbert-Stein. We prove this fact for the following kinds of bounded domains of holomorphyD in C ⁿ : 1)n=1 and ₁(D) is finitely generated, 2) the group of automorphisms ofD in the compactopen topology is compact, 3)D is strongly pseudoconvex with smooth boundary andn2, 4)D is symmetric of type I.     

Spaces of holomorphic functions in regular domains

Let Ω be a regular domain in the complex plane , . Let be the linear space over of the holomorphic functions f in Ω such that f⁽ⁿ⁾ is bounded in Ω and is continuously extendible to the closure of Ω, n=0,1,2,… . We endow , in a natural manner, with a structure of Fréchet space and we obtain dense subspaces F of , with good topological linear properties, also satisfying that each function f of F, distinct from zero, does not extend holomorphically outside Ω.