Factorisations of the Helmholtz Operator, Radó’s Theorem, and Clifford Analysis

Radó’s theorem for holomorphic functions asserts that if a continuous function is holomorphic on the complement of its zero locus, then it is holomorphic everywhere. We prove in this paper an equivalent theorem for functions lying in the kernel of a first order differential operator D{\mathcal{D}} such that the Helmholtz operator ∇²+λ can be factorized as the composition [^(D)]D{\widehat{\mathcal{D}}\mathcal{D}} . We also analyse the factorisations [^(D)]D{\widehat{\mathcal{D}}\mathcal{D}} of the Laplace and Helmholtz operators associated to the Clifford analysis and the representations of holomorphic function of several complex variables.     

Über das Randverhalten von holomorphen Automorphismen beschränkter Gebiete

Let D Cⁿ be a bounded positive (p₁,...,p_n)-circular domain which contains the origin. We prove: Every holomorphic automorphism g of D can be extended holomorphically to a neighbourhood of ¯D. In part II we prove the same statement for a certain class of Rheinhardt domains.     

与例外函数和分担函数相关的亚纯函数的正规族

设F为区域D内的一族亚纯函数,对于每个f∈F,f的所有极点重数至少是2,a(z)和b(z)为两个在D内满足a(z)■b(z)的全纯函数.若对于每个f∈F,f(z)≠a(z)和f(z)≠b(z),则F在D内正规.这个结果改进了经典的Montel定则.此外,我们也讨论了亚纯函数族中每个函数与其导函数分担两个全纯函数的正规性.     

Representing systems for the space of holomorphic functions in a (ρ,α)-convex domain

This article is a natural supplement to Chapter 5 ofAsymptotic Characteristics of Entire Functions and Their Applications [Nauka, Novosibirsk (1991)]. Earlier, for the functions holomorphic in the closure of a (ρ,α)-convex bounded domain D, the author found a criterion of the existence of a series expansion in a special system of entire functions. Here it is shown that the same criterion applies to the functions holomorphic in D. Moreover, a new integral representation of entire functions is described, which enables one to construct new representing systems for the spaces of holomorphic functions and H(D). Bibliography: 17 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 206, 1993, pp. 91–196. Translated by D. V. Yakubowich.     

Sur l'extension des fonctions CR dans une variété de stein

Sumé Etant donné un domaine D relativement compact d'une variété de Stein M de dimension n, n 2, on montre que toute fonction continue, CR, définie sur un ouvert connexe de D ayant un complémentaire K dont l'enveloppe holomorphiquement convexe dans M ne rencontre pas ¯ DK, se prolonge en une fonction holomorphe sur D.

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Summary Let there be given a relatively compact domain D in a Stein manifold M of dimension n, n 2, we prove the holomorphic extendibility of the continuous CR functions defined on an open connected subset of D, provided theO(M)-hull of its complementary K does not meet ¯ DK.

     

Normality concerning the sequence of functions

Let {f_n} be a sequence of functions meromorphic in a domain D, let {h_n} be a sequence of holomorphic functions in D, such that that h(z)→h(z), where h.(z)→0 is holomorphic in D, and let k be a positive integer. If for each n∈N~+, f_n(z)≠0 and f_n~(k)(z)-h_n(z) has at most k distinct zeros(ignoring multiplicity) in D, then {f_n} is normal in D.     

Montel正规定则的推广(英文)

用简单的方法证明了全纯函数族的一个正规定则,推广了Montel正规定则.设F为区域D上的一个全纯函数族,其零点是重级的,a为有限复数.如果f,g∈F,有(f-1)f′与(g-1)g′分担a,则F正规.     

Permissible growth rates for Birkhoff type universal harmonic functions

A harmonic function H on is said to be universal (in the sense of Birkhoff) if its set of translates is dense in the space of all harmonic functions on with the topology of local uniform convergence. The main theorem includes the result that such functions, H, can have any prescribed order and type. The growth result is compared with a similar known theorem for G.D. Birkhoff's universal holomorphic functions and contrasted with known growth theorems for MacLane-type universal harmonic and holomorphic functions.     

The representation of regular functions by Dirichlet series

It is proved that if a system of exponents has the property that any function regular in a closed convex domain ¯G can be represented in an open domain G by a Dirichlet series, then any function regular only in G can be represented in G by a Dirichlet series with the same system of exponents. A study is made of the representation of functions regular in ¯G by Dirichlet series that converge in ¯G.Translated from Matematicheskie Zametki, Vol. 21, No. 5, pp. 641–651, May, 1977.     

The Bergman kernel function

In this note, we point out that a large family of n×n matrix valued kernel functions defined on the unit disc $ \mathbb{D} \subseteq \mathbb{C} $ \mathbb{D} \subseteq \mathbb{C} , which were constructed recently in [9], behave like the familiar Bergman kernel function on $ \mathbb{D} $ \mathbb{D} in several different ways. We show that a number of questions involving the multiplication operator on the corresponding Hilbert space of holomorphic functions on $ \mathbb{D} $ \mathbb{D} can be answered using this likeness.     

Bergman-Hartogs型域的全纯自同构群

我们考虑一类以有界对称域D为底的Bergman-Hartogs型域Ω={(wm(1),...,w(r),z)∈C1×···×Cmr×D:∥w(1)∥2p1+···+∥w(r)∥2prKD(z,z)-q},其中KD(z,z)是D上的Bergman核函数,r 1且为正整数,参数p1,...,pr1和q0为实数.我们给出它的全纯自同构群,并且证明当r=1时此自同构群为最大全纯自同构群;当r1时,若Ω的全纯自同构变换F将(0,z)∈{0}×D映到(0,z*)∈{0}×D,则F在我们给出的全纯自同构群中.     

Schwarz-Pick estimates for bounded holomorphic functions in the unit ball of ℂ n

We give a Schwarz-Pick estimate for bounded holomorphic functions on unit ball in Cn, and generalize some early work of Schwarz-Pick estimates for bounded holomorphic functions on unit disk in C.     

单位球上全纯函数的高阶Schwarz-Pick估计*

作者给出了单位球B_n ? Cⁿ到凸区域?? C上全纯函数的高阶Schwarz-Pick估计. 通过引入双曲度量,得到了单位圆盘D到凸区域?上全纯函数的系数估计. 应用该系数估计结果,得到单位球B_n到?内的全纯函数的高阶Schwarz-Pick估计.特别地, 当?是单位圆盘或右半平面时, 得到的结果分别与熟知的结果是一致的.     

Interpolation of bounded sequences

This paper deals with an interpolation problem in the open unit disc ⅅ of the complex plane. We characterize the sequences in a Stolz angle of ⅅ, verifying that the bounded sequences are interpolated on them by a certain class of not bounded holomorphic functions on ⅅ, but very close to the bounded ones. We prove that these interpolating sequences are also uniformly separated, as in the case of the interpolation by bounded holomorphic functions.     

关于全纯函数的正规定则

研究涉及微分多项式与公共值的全纯函数族的正规性问题.设F为区域D内的全纯函数族,n 1为任一正整数,b为有限常数,如果对F中任意两个函数f与g,fn(f-1)f′与gn(g-1)g′在D内都以b为公共值,则F在D内正规.此结果对亚纯函数不成立.     

On the nonextendability of germs of holomorphic functions

We construct explicit supporting manifolds and local holomorphic peak functions as obstructions to the extendability of holomorphic functions on a class of domains not necessarily pseudoconvex in C^N, N >2.     

关于杨乐及Schwick的一结果

设ψ■0为复平面区域D内的只有单零点的全纯函数,k为正整数,F为区域D内的亚纯函数族.如果每个f∈F满足f≠0且只有重极点;对F内任一组函数f与g,f(k)与g(k)在D内分担ψ(z),则F在D内正规.     

Boundary Distortion Estimates for Holomorphic Maps

We establish some estimates of the angular derivatives from below for holomorphic self-maps of the unit disk ${\mathbb {D}}$ at one and two fixed points of the unit circle provided there is no fixed point inside ${\mathbb {D}}$ . The results complement Cowen–Pommerenke and Anderson–Vasil’ev type estimates in the case of univalent functions. We use the method of extremal length and a semigroup approach to deriving inequalities for holomorphic self-maps of the disk which are not necessarily univalent using known inequalities for univalent functions. This approach allowed us to obtain a new Ossermans type estimate as well as inequalities for holomorphic self-maps which images do not separate the origin and the boundary.     

Schwarz-Pick estimates for positive real part holomorphic functions on unit ball and polydisc

We give higher order derivatives Schwarz-Pick estimates for all positive real part holomorphic functions on Bn and D n,and generalize early work on Schwarz-Pick estimate of higher order derivatives for holomorphic functions with positive real part on unit disk in C.     

Integral representation with weights II, division and interpolation

Let f be a r×m-matrix of holomorphic functions that is generically surjective. We provide explicit integral representation of holomorphic ψ such that ϕ=f ψ, provided that ϕ is holomorphic and annihilates a certain residue current with support on the set where f is not surjective. We also consider formulas for interpolation. As applications we obtain generalizations of various results previously known for the case r=1. The author was partially supported by the Swedish Research Council