Two Elementary Analytic Functions and Their Relationship with Hardy and Bergman Spaces (original) (raw)

On Generators of the Hardy and the Bergman Spaces

arXiv (Cornell University), 2023

A function ϕ which is analytic and bounded in the unit disk D is called a generator for the Hardy space H 2 (D) or the Bergman space A 2 (D) if polynomials in ϕ are dense in the corresponding space. We characterize generators in terms of ϕ−invariant subspaces which are also z−invariant and study wandering properties of such subspaces. Density of bounded analytic functions in the ϕ−invariant subspaces of H 2 (D) is also investigated.

Spaces of analytic functions of Hardy-Bloch type

Journal D Analyse Mathematique, 2006

For 0 < p ≤ ∞ and 0 < q ≤ ∞, the space of Hardy-Bloch type B(p, q) consists of those functions f which are analytic in the unit disk D such that (1 − r)Mp(r, f ) ∈ L q (dr/(1 − r)). We note that B(∞, ∞) coincides with the Bloch space B and that B ⊂ B(p, ∞), for all p. Also, the space B(p, p) is the Dirichlet space D p p−1 . We prove a number of results on decomposition of spaces with logarithmic weights which allow us to obtain sharp results about the mean growth of the B(p, q)-functions. In particular, we prove that if f is an analytic function in D and 2 ≤ p < ∞, then the condition Mp(r, f ) = O (1 − r) −1 ¡ , as r → 1, implies that Mp(r, f ) = O log 1 1−r 1/2 , as r → 1. This result is an improvement of the well known estimate of Clunie and MacGregor and Makarov about the integral means of Bloch functions, and it also improves the main result in a recent paper by Girela and Peláez. We also consider the question of characterizing the univalent functions in the spaces B(p, 2), 0 < p < ∞, and in some other related spaces and give some applications of our estimates to study the Carleson measures for the spaces B(p, 2) and D p p−1 . D |f (z)| p dA(z) < ∞ 2000 Mathematics Subject Classification. 30D45, 30D55.

New Characterizations of Bergman Spaces

Ann. Acad. Sci. Fenn. Math, 2008

We obtain several new characterizations for the standard weighted Bergman spaces A p α on the unit ball of C n in terms of the radial derivative, the holomorphic gradient, and the invariant gradient.

Holomorphic N K and Bergman-type spaces

Birkh¨auser Verlarg Publisher Basel/Switzerland, 2008

In this paper we introduce a new class of functions, called NK-type space of analytic functions by the help of a nondecreasing function K : [0, ∞) → [0, ∞). Further, under mild conditions on the weight function K we characterize lacunary series in NK space. Finally, we study the boundedness and compactness of composition operators between NK and Bergman spaces. Mathematics Subject Classification (2000). Primary 47B33; 47B38 Secondary 30H05.

Bergman and Reinhardt weighted spaces of holomorphic functions

Illinois Journal of Mathematics

We study isometries between spaces of weighted holomorphic functions defined on bounded domains in C n. Using the Bergman kernel we see that it is possible to define a 'natural' weight on bounded domains in C n. We calculate the isometries of weighted spaces of holomorphic functions on the unit ball, the Thullen domains, the generalised Thullen domains and the domain with minimal complex norm.

Boundary behaviour of analytic functions in spaces of Dirichlet type

Journal of Inequalities and Applications, 2006

For 0 < p < ∞ and α > −1, we let Ᏸ p α be the space of all analytic functions f in D = {z ∈ C : |z| < 1} such that f belongs to the weighted Bergman space A p α . We obtain a number of sharp results concerning the existence of tangential limits for functions in the spaces Ᏸ p α . We also study the size of the exceptional set E

On the zeros of functions in Bergman spaces and in some other related classes of functions

Journal of Mathematical Analysis and Applications, 2005

A well-known theorem of H.S. Shapiro and A.L. Shields implies that if f ≡ 0 is a function which belongs to the Bergman space A p (0 < p < ∞) and {z k } is a sequence of zeros of f which is contained in a Stolz angle, then {z k } satisfies the Blaschke condition. In this paper we improve this result. We consider a large class of regions contained in the unit disc D which touch ∂D at a point ξ tangentially and we prove that the mentioned result remains true if we substitute a Stolz angle by any of these regions of tangential approach.  2004 Elsevier Inc. All rights reserved.

Generaliz Generalization of the H Tion of the Hardy Class for an Ass for Analytic Functions

Scientific reports of Bukhara state University, 2021

Introduction. Quoting from a well-known American mathematician Lipman Bers [1]-It would be tempting to rewrite history and to claim that quasiconformal transformations have been discovered in connection with gas-dynamical problems. As a matter of fact, however, the concept of quasiconformality was arrived at by Grotzsch [2] and Ahlfors [3] from the point of view of function theory‖. The present work is devoted to the theory of analytic solutions of the Beltrami equation () () (), (1) z z f z A z f z  which directly related to the quasi-conformal mappings. The function () Az is, in general, assumed to be measurable with | () | 1 A z C  almost everywhere in the domain D  under consideration. Solutions of equation (1) are often referred to as () Az  analytic functions in the literature. Research methods. The solutions of equation (1), as well as quasi-conformal homeomorphisms in the complex plane have been studied in suffient details. The purpose of this paper is to study () Az  analytic functions in a particular case, when the function () Az is anti-holomorphic in a considered domain [19]. As we can see below, in this spesial case the solution of (1) possesses many properties of analytic functions, has an integral in the norm is a function of the Hardy class and this class is generalized. Results and discussions. The aim of this paper is to investigate () Az  analytic functions in special case when the function () Az is an anti-analytic function in a domain. Also, in paper introduces some classes for () Az  analytic functions. Nevanlinn's theorem for () Az  analytic functions is proved and its results are given. Examples of functions belonging to these classes in different cases are given. The theorems of Riesz and Smirnov for () Az  analytic functions are proved. Conclusion. The theory of boundary properties made considerable advances in the first third of the 20th century, owing to the work of several scientists; it resumed its rapid advance in the second half of that century, accompanied by the appearance of new ideas and methods, novel directions and objects of study. Its development is closely connected with various fields of mathematical analysis and mathematics in general, first and foremost with probability theory, the theory of harmonic functions, the theory of conformal mapping, boundary value problems of analytic function EXACT AND NATURAL SCIENCES 30 SCIENTIFIC REPORTS OF BUKHARA STATE UNIVERSITY 2021/4 (86) theory. The theory of boundary properties of analytic functions is closely connected with various fields of application of mathematics by way of boundary value problems. The theory of boundary properties of analytical functions, which grew out of the