Set Of Holomorphic Functions Research Articles

Estimates of Level Sets of Holomorphic Functions and Applications to the Weighted Log Canonical Thresholds

In this paper, we establish some estimates of level sets of holomorphic functions. Relying on obtained estimates we compute some of the weighted log canonical thresholds of plurisubharmonic functions. Finally, we prove the analyticity of the sublevel sets of weighted log canonical thresholds of plurisubharmonic functions.

Iteration of Holomorphic Function Systems on the Riemann Sphere

In this work, it is shown that a set of holomorphic functions, normal in any set, remains normal in the same set under the action of another holomorphic function. And therefore, we verify that Julia's sets of two-function composition (no matter the order of composition) coincide, up to a rescaling.

Radial limit zero sets of holomorphic functions on the ball

We extend to higher dimensions a theorem of Lusin and Privalov concerning radial limit zero sets of holomorphic functions on the unit disc, and we show that, in contrast to the case in dimension one, the converse fails for holomorphic functions on the unit ball in . Finally, we show that in higher dimensions any characterization of the radial limit zero sets of holomorphic functions on the ball must take into account the ‘complex structure’ on the sphere.

On Blaschke products with finite Dirichlet-type integral

The class of functions with finite Dirichlet-type integral is defined as the set of holomorphic functions f in the unit disk satisfying the following condition: $$\int_0^{2\pi } {\int_0^1 {(1 - r)^\alpha \left| {f'(re^{i\theta } )} \right|^p r dr d\theta , \alpha > - 1, 0 < p < + \infty } .}$$ These classes are usually denoted by Dαp. In this paper, we prove the converse of Rudin’s theorem and thus provide a necessary and sufficient condition for a Blaschke product to belong to the class D01.

Spaceability and algebrability in the theory of domains of existence in Banach spaces

We show that if \(U\) is a connected domain of existence in a separable Banach space, then the set of holomorphic functions on \(U\) whose domain of existence is \(U\) is closely strongly algebrable and densely strongly algebrable.

Fivebrane instantons in Calabi-Yau compactifications

We provide the last missing piece of the complete non-perturbative description of the low energy effective action emerging from Calabi-Yau compactifications of type II string theory --- NS5-brane instanton corrections to the hypermultiplet moduli space $M_H$. We find them using S-duality symmetry of the type IIB formulation. The result is encoded in a set of holomorphic functions on the twistor space of $M_H$ and includes all orders of the instanton expansion.

Domains of convergence for monomial expansions of holomorphic functions in infinitely many variables

Let ℱ(R) be a set of holomorphic functions on a Reinhardt domain R in a Banach sequence space (as e.g. all holomorphic functions or all m-homogeneous polynomials on the open unit ball of ). We give a systematic study of the sets dom ℱ(R) of all z ∈ R for which the monomial expansion of every ƒ ∈ ℱ(R) converges. Our results are based on and improve the former work of Bohr, Dineen, Lempert, Matos and Ryan. In particular, we show that up to any ε > 0 is the unique Banach sequence space X for which the monomial expansion of each holomorphic function ƒ converges at each point of a given Reinhardt domain in X. Our study shows clearly why Hilbert's point of view to develop a theory of infinite dimensional complex analysis based on the concept of monomial expansion, had to be abandoned early in the development of the theory.

Hardy-Sobolev spaces and algebras of holomorphic functions on the unit ball in ℂ n

For 0 < p ≤ ∞ let H p (𝔹 n ) denote the usual Hardy space of holomorphic functions on the unit ball 𝔹 n in ℂ n , n ≥ 2. If f is a holomorphic function on 𝔹 n , the radial derivative ℛ f is defined by and for m = 2, 3, … , the mth order radial derivative ℛ m is defined by ℛ m f = ℛ(ℛ m − 1 f). The Hardy-Sobolev space of order m is defined as the set of holomorphic functions f on 𝔹 n for which ℛ m f ∈ H p (𝔹 n ). The main result is as follows: Theorem: Let m ∈ {1, 2, 3, …}. If a. m ≥ n and , or b. 1 ≤ m < n (n ≥ 2) and , then is an algebra. Furthermore, ∥ ∥ p,m,λ defined for p ≥ 1 by where is a sequence of positive numbers satisfying is a Banach algebra norm on whenever p, m with p ≥ 1 satisfy one of (a) or (b) above. In the article we also prove that if m > n and , then is a p-Banach algebra under a suitable p-norm on . By examples it is shown that the results are best possible.

On extreme points of some sets of holomorphic functions

In the Banach space H∞ (U) of bounded, holomorphic functions in the unit disc U of the complex plane, the paper considers the subset H∞ (U; K) of functions that take values in a compact K. A description of the extreme points of H∞ (U; K) is given under rather general conditions on K.

A Montel Theorem for Holomorphic Functions on Infinite Dimensional Spaces that Omit the Values 0 and 1

Let X be a connected complex Banach manifold. We give a proof that the set of holomorphic functions on X that omit the values 0 and 1 is a normal family.

Description of some weighted spaces of holomorphic functions in terms of fractional derivatives

Assuming that Un is the unit polydisk in the n-dimensional complex plane , H(U n ) is the set of holomorphic functions in U n , S is the space of functions of regular variation and ω =(ω1, …,ω n ), ω j ∈S, by Lp (ω) we denote the class of all measurable functions defined in Un and such that where dm 2n (z) is the 2n-dimensional Lebesgue measure in U n . In this article, for any p∈(0,∞) we prove that if and only if , where D β stands for the fractional derivative of f of some multi-index order β.

Monodromy-data parameterization of spaces of local solutions of integrable reductions of Einstein’s field equations

We show that for the fields depending on only two of the four space-time coordinates, the spaces of local solutions of various integrable reductions of Einstein’s field equations are the subspaces of the spaces of local solutions of the “null-curvature” equations selected by universal (i.e., solution-independent conditions imposed on the canonical (Jordan) forms of the desired matrix variables. Each of these spaces of solutions can be parameterized by a finite set of holomorphic functions of the spectral parameter, which can be interpreted as a complete set of the monodromy data on the spectral plane of the fundamental solutions of associated linear systems. We show that both the direct and inverse problems of such a map, i.e., the problem of finding the monodromy data for any local solution of the null-curvature equations for the given Jordan forms and also of proving the existence and uniqueness of such a solution for arbitrary monodromy data, can be solved unambiguously (the “monodromy transform”). We derive the linear singular integral equations solving the inverse problem and determine the explicit forms of the monodromy data corresponding to the spaces of solutions of Einstein’s field equations.

CONTACT ZONE MODELS FOR AN INTERFACE CRACK IN A THERMOMECHANICALLY LOADED ANISOTROPIC BIMATERIAL

An interface crack between two anisotropic semi-infinite spaces under the action of remote mixed-mode mechanical loading and a temperature flux is considered. Assuming that all fields are independent of the x 3 -coordinate co-directed with the crack front, the stresses and the temperature flux as well as the jumps of the displacements and the temperature at the interface are presented via a set of holomorphic functions in the whole (x 1 , x 2 )-plane with a cut along the crack area. By means of this representation a solution for an open crack model can be given in an analytical form, and further an inhomogeneous combined Dirichlet-Riemann boundary value problem could be formulated for a crack with an artificial contact zone. An exact analytical solution of this problem has been found, and the stress intensity factors are presented for different contact zone lengths with a special success in the case of a small length of the contact zone. Furthermore, it is shown that the obtained solution can be used for the numerical solution of interface crack problems for finite-sized bimaterials. Real contact zone lengths and the associated stress intensity factors are found from the obtained solution; in addition, their dependencies on the intensity of the temperature flux and the mechanical load mixity are demonstrated.

Peak sets in real-analytic convex boundaries

Throughtout this paper we will denote by D a smoothly bounded domain in C with defining function r. We denote by A∞(D) the set of holomorphic functions in D which have C∞-extension to D. First, we give the necessary definitions and notations needed in this paper. A closed subset K ⊂ ∂D is a peak set for A∞(D) if there exists a function f ∈ A∞(D) so that f = 1 on K and |f | 0 on D \K. Such a function g is called a strong support function for K. We denote by Tp(M) the real tangent space to a smooth manifold M at the point p ∈ M. For a point p ∈ M , the complex tangent space of M at p denoted by T C p (M) is the maximal complex subspace of Tp(M), of complex dimension n− 1 if M = ∂D. A C∞-submanifold M ⊆ ∂D is integral at p ∈M if Tp(M) ⊆ T C p (∂D). M is an integral manifold if it is integral at each point p ∈M. A C∞-submanifold M ⊂ ∂D is totally real if T C p (M) = {0} for every p ∈M. We denote by w(∂D) the set of weakly pseudoconvex boundary points. For p ∈ ∂D, we let Np denote the null space in T C p (∂D) of the Levi form at

Hypercomplex Algebraic Geometry

It is well-known that sums and products of holomorphic functions are holomorphic, and the holomorphic functions on a complex manifold form a commutative algebra over C . The study of complex manifolds using algebras of holomorphic functions upon them is called complex algebraic geometry. The purpose of this paper is to develop an analogue of complex algebraic geometry, in which the complex numbers C are replaced by the quaternions H . The natural quaternionic analogue of a complex manifold is called a hypercomplex manifold. A class of H -valued functions on hypercomplex manifolds will be defined, called q-holomorphic functions, that are analogues of holomorphic functions on complex manifolds. Now, the set of holomorphic functions on a complex manifold is a commutative algebra over C . Therefore one asks: does the set of q-holomorphic functions on a hypercomplex manifold have an analogous algebraic structure, and if so, what is it? We shall show that the q-holomorphic functions on a (noncompact) hypercomplex manifold do indeed possess a rich algebraic structure. To describe it, we shall introduce a theory of quaternionic algebra, which is a quaternionic analogue of real linear algebra. This theory is built on three building blocks: AH-modules, the analogues of vector spaces, AH-morphisms, the analogues of linear maps, and the quaternionic tensor product, the analogue of tensor product of real vector spaces. As far as the author can tell, these ideas seem to be new. They enable us to construct algebraic structures over H as though H were a commutative field. Quaternionic algebra describes the algebraic structure of hypercomplex manifolds in a remarkable way, and it seems to be the natural language of hypercomplex algebraic geometry, the algebraic geometry of hypercomplex manifolds. We believe that quaternionic algebra is worth studying for its own sake. It has many similarities with linear algebra over R or C , which is why the analogies between complex and quaternionic theories work so well, but there are also deep differences, which give quaternionic algebra a flavour all of its own. Quillen [12] has given a sheaf-theoretic interpretation of the ideas of quaternionic algebra, based on a previous version of this paper. He finds a contravariant equivalence between a class of AH-modules and regular sheaves on a real form of CP. Regular sheaves on CP are equivalent to representations of the Kronecker quiver, and out of such a representation Quillen constructs an AH-module. Under Quillen’s equivalence stable AH-modules correspond to regular vector bundles over the real form of CP.

Compact Sets of Holomorphic Mappings

Compactness of locally bounded sets of holomorphic functions with infinite dimensional domains is connected, using Heinrich's density condition, to the Schwartz and semi-Montel properties on the domain. The metrizability of bounded subsets for various spaces of holomorphic functions is investigated.

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.

Division and extension in weighted Bergman-Sobolev spaces

Let D be a bounded strictly pseudoconvex domain of Cn with C 8 boundary and Y = {z; u1(z) = ... = ul(z) = 0} a holomorphic submanifold in the neighbourhood of D', of codimension l and transversal to the boundary of D. In this work we give a decomposition formula f = u1f1 + ... + ulfl for functions f of the Bergman-Sobolev space vanishing on M = Y n D. Also we give necessary and sufficient conditions on a set of holomorphic functions {fa}|a|=m on M, so that there exists a holomorphic function in the Bergman-Sobolev space such that Daf |M = fa for all |a| = m.

Tangential boundary limits and exceptional sets for holomorphic functions in Dirichlet-type spaces

According to the well-known theorem of Fatou, the harmonic extension u to the upper half-space R~_ +x of a function f in LP(Rn), 1 1. In this paper we study the analogous question in the unit ball B of C'. Recall that the correct analogue of Fatou's theorem for B is the theorem of Kor/myi [K] stating that invariant harmonic extensions (Poisson-Szeg6 integrals) of functions in LP(S), 1 < p < oo, have "admissible" limits at almost every point of the unit sphere S. In particular, this is true for holomorphic functions in the Hardy space HP(B). As is well known, Korfinyi's admissible regions allow parabolic tangential approach along the complex-tangential directions. These admissible regions are best possible as far as the order of tangency to the boundary is concerned, even if only bounded holomorphic functions are considered, as shown in [H-S] (however, they are not best possible in a stronger sense [Su 1]). By analogy with the real-variable situation, it seems reasonable to expect Kor/myi's regions not to be optimal for holomorphic functions whose (admissible) boundary values are, in some sense, Bessel-type potentials of LP(S) functions. Thus we consider holomorphic functions in the unit ball which are given by "fractional Cauchy integrals" of LP(S) functions. These are obtained by modifying the exponent in the denominator of the Cauchy kernel of B, a procedure suggested by the

Removable sets for holomorphic functions of several complex variables

We show that every closed subset of CN that has finite (2N - 2)-dimensional measure is a removable set for holomorphic functions, and we obtain a related result on the ball.