Bergman-type Spaces Research Articles

Small Hankel operators between Bergman-type spaces in the unit ball

We utilize the ( A α p , q ) -Carleson measure to investigate the bounded small Hankel operator h f α acting between the Bergman-type space A α p and A α q ¯ on the complex unit ball B n for some α ≤ − 1 . For n ≥ 2 , and f ∈ H ( B n ) , we have the conclusions: Suppose α ≤ − 1 , max { 1 , − 2 α − 2 } < p < ∞ and -1 $ ]]> pN + α > − 1 . Then h f α : A α p → A α p ¯ is bounded if and only if | R N f ( z ) | p d v pN + α ( z ) is an ( A α p , p ) -Carleson measure. Suppose − n − 1 < α ≤ − 1 , max { 1 , − 2 α − 2 } < p < q < ∞ and \\max\\left\\{ \\frac {q-p}p (n+1+\\alpha)-2(\\alpha+1),\\frac{-p(\\alpha+1)}{p-1}\\right\\}. \\end{align*} $$]]> pN > max { q − p p ( n + 1 + α ) − 2 ( α + 1 ) , − p ( α + 1 ) p − 1 } . Then h f α : A α p → A α q ¯ is bounded if and only if | R N f ( z ) | q d v qN + α ( z ) is an ( A α p , q ) -Carleson measure. Suppose 1 < q < p < ∞ , α ≤ − 1 , N be a positive integer such that -1 $ ]]> qN + α > − 1 . If | R N f ( z ) | q d v qN + α ( z ) is an ( A α p , q ) -Carleson measure, then h f α : A α p → A α q ¯ is bounded. Let κ = pq p − q and κ ′ = κ κ − 1 . If -1 $ ]]> α + ( p − 1 ) κ ′ N > − 1 and h f α : A α p → A α q ¯ is bounded, then f ∈ A α κ with ‖ f ‖ A α κ ≲ ‖ h f α ‖ . If | R N f ( z ) | q d v qN + α ( z ) is an ( A α p , q ) -Carleson measure, then f ∈ A α κ . But, there is an f ∈ A α κ such that | R N f ( z ) | q d v qN + α ( z ) is not an ( A α p , q ) -Carleson measure.

Differences weighted composition operators acting between kind of weighted Bergman-type spaces and the Bers-type space -I-

&lt;abstract&gt;&lt;p&gt;Let $ \mathcal{O}(\mathbb{D}) $ denote the class of all analytic or holomorphic functions on the open unit disk $ \mathbb{D} $ of $ \mathbb{C} $. Let $ \varphi $ and $ \psi $ are an analytic self-maps of $ \mathbb{D} $ and $ u, v\in \mathcal{O}(\mathbb{D}). $ The difference of two weighted composition operators is defined by&lt;/p&gt; &lt;p&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE1"&gt; \begin{document}$ T_{\varphi, \psi}f(z): = \bigg(W_{\varphi, \;u}f- W_{\psi, \;v}f\bigg)(z) = u (z)(f\circ \varphi)(z) -v(z)(f\circ \psi)(z), \ f \in \mathcal{O}(\mathbb{D})\;\hbox{and}\;z\in \mathbb{D}. $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt; &lt;p&gt;The boundedness and compactness of the differences of two weighted composition operators from $ {\cal H}_{\alpha}^\infty(\mathbb{D}) $ spaces into $ \mathcal{N}_{K}(\mathbb{D}) $ spaces (resp. from $ \mathcal{N}_{K}(\mathbb{D}) $ into $ {\cal H}_{\alpha}^\infty (\mathbb{D}) $) are investigate in this paper.&lt;/p&gt;&lt;/abstract&gt;

Affine density, von Neumann dimension and a problem of Perelomov

We provide a solution to Perelomov's 1972 problem concerning the existence of a phase transition (known in signal analysis as ‘Nyquist rate’) determining the basis properties of certain affine coherent states labelled by Fuchsian groups. As suggested by Perelomov, the transition is given according to the hyperbolic volume of the fundamental region. The solution is a more general form (in phase space) of the PSL(2,R) variant of a 1989 conjecture of Kristian Seip about wavelet frames, where the same value of ‘Nyquist rate’ is obtained as the trace of a certain localization operator. The proof consists of first connecting the problem to the theory of von Neumann algebras, by introducing a new class of projective representations of PSL(2,R) acting on non-analytic Bergman-type spaces. We then adapt to this setting a new method for computing von Neumann dimensions, due to Sir Vaughan Jones. Our solution contains necessary conditions in the form of a ‘Nyquist rate’ dividing frames from Riesz sequences of coherent states and sampling from interpolating sequences. They hold for an infinite sequence of spaces of polyanalytic functions containing the eigenspaces of the Maass operator and their orthogonal sums. Within mild boundaries, we show that our result is best possible, by characterizing our sequence of function spaces as the only invariant spaces under the non-analytic PSL(2,R)-representations.

On Bergman spaces induced by a v-Laplacian vector fields theory

The history of Bergman spaces goes back to the book [4] in the early fifties by S. Bergman, where the first systematic treatment of the subject was given, and since then there have been a lot of papers devoted to this area. Some standard works here are [5,8,15,28] and the references therein, which contain a broad summary and historical notes of the subject, that frees us from referring to missing details. In their recent works González-Cervantes, Luna-Elizarrarás and Shapiro [11,12], laid the foundations for the generalization of the theory of Bergman spaces induced by Laplacian (sometimes called solenoidal and irrotational, or harmonic) vector fields by taking advantage on the intimate connections between harmonic vector fields theory and quaternionic analysis for the Moisil-Theodorescu operator (MT-operator for short). A deeper discussion of the last mentioned relation can be found in [1]. On the setting of general bounded domains in R3, we extend the aforementioned study in a very natural way to the case of an introduced v-MT-operator for v∈R3, proving several properties of induced Bergman spaces and some relative results about Stokes and Borel-Pompieu formulas for v-MT-hyperholomorphic functions, i.e., functions which belong to kernel of the v-MT-operator. In addition, we show that this v-MT operator satisfies a conformal co-variant property. Application of all the above allows to study of Bergman type spaces induced by v-Laplacian vector fields theory, which represents the main goal of this paper.

$${\cal N}\left( {p,q,s} \right)$$-Type Spaces in the Unit Ball of ℂn. IV: Atomic Decomposition, Gleason’s Problem and Distance Problems

The purpose of this paper is to study several different properties of the holomorphic $${\cal N}\left( {p,q,s} \right)$$ -type functions in the unit ball $$\mathbb{B}$$ of ℂn via Carleson measure techniques. More precisely, we establish an atomic decomposition on such spaces and then we solve the Gleason’s problem on them. We also study the distance problems between Bergman-type spaces $${A^{ - {q \over p}}}\left(\mathbb{R} \right)$$ and $${\cal N}\left( {p,q,s} \right)$$ -type spaces. These results yield some new characterizations of the holomorphic F(p, q, t)-functions.

On some new sharp embedding theorems for multifunctional Herz-type and Bergman-type spaces in tubular domains over symmetric cones

We introduce new multifunctional mixed norm analytic Herz-type spaces in tubular domains over symmetric cones and provide new sharp embedding theorems for them. Some results are new even in case of onefunctional holomorphic spaces. Some new related sharp results for new multifunctional Bergman-type spaces will be also provided under one condition on Bergman kernel.

Weighted composition operators on weighted Bergman spaces induced by doubling weights

In this paper, we investigate the boundedness, compactness, essential norm and the Schatten class of weighted composition operators $uC_\varphi $ on Bergman type spaces $A_\omega ^p $ induced by a doubling weight ω. Let $X=\{u\in H(\mathbb{D} ): uC_\varphi \colon A_\omega ^p\to A_\omega ^p\ \text {is bounded}\}$. For some regular weights ω, we obtain that $X=H^\infty $ if and only if ϕ is a finite Blaschke product.

On properties of solutions to the α-harmonic equation

The aim of this paper is to establish properties of solutions to the α-harmonic equations: , where is a continuous function and denotes the closure of the unit disc in the complex plane . We obtain Schwarz type and Schwarz-Pick type inequalities for solutions to the α-harmonic equation. In particular, for , solutions to the above equation are called α-harmonic functions. We determine the necessary and sufficient conditions for an analytic function ψ to have the property that is α-harmonic function for any α-harmonic function f. Furthermore, we discuss the Bergman-type spaces on α-harmonic functions.

On some new sharp embedding theorems for multifunctional Herz-type and Bergman-type spaces in pseudoconvex domains

We introduce new multifunctional mixed norm analytic Herz-type spaces in strongly pseudoconvex domains and provide new sharp embedding theorems for them. Some results are new even in case of onefunctional holomorphic spaces. Some new related sharp results for new multifunctional Bergman-type spaces will be also provided under one condition on Bergman kernel. Similar results with similar proofs in unbounded tubular domains over symmetric cones and bounded symmetric domains will be also shortly mentioned.

On mixed norm Bergman–Orlicz–Morrey spaces

Abstract Following the ideas of our previous research, in this paper we continue the study of new Bergman-type spaces on the unit disc with mixed norm in terms of Fourier coefficients. Here we deal with the case where the sequence of norms of Fourier coefficients in the Orlicz–Morrey space in radial variable belongs to l q {l^{q}} . We study the boundedness of the Bergman projection and provide a description of functions in these spaces via the behavior of their Taylor coefficients.

Boundedness of the Bergman Projection and Some Properties of Bergman Type Spaces

We give a simple proof of the boundedness of Bergman projection in various Banach spaces of functions on the unit disc in the complex plain. The approach of the paper is based on the idea of Zaharyuta and Yudovich (Uspekhi Mat Nauk 19(2):139–142, 1964) where the boundedness of the Bergman projection in Lebesgue spaces was proved using Calderon–Zygmund operators. We exploit this approach and treat the cases of variable exponent Lebesgue space, Orlicz space and variable exponent generalized Morrey spaces. In the case of variable exponent Lebesgue space the boundedness result is known, so in that case we provide a simpler proof, whereas the other cases are new. The major idea of this paper is to show that the approach can be applied to a wide range of function spaces. We also study the rate of growth of functions near the boundary in spaces under consideration and their approximation by mollified dilations.

On Bergman type spaces of holomorphic functions and the density, in these spaces, of certain classes of singular functions

We consider Bergman spaces and variations of them on domains in one or several complex variables. For certain domains , we show that the generic function in these spaces is totally unbounded in and hence non-extendable. We also show that generically these functions do not belong – not even locally – to Bergman spaces of higher order. Finally, in certain domains , we give examples of bounded non-extendable holomorphic functions – a generic result in the spaces of holomorphic functions in whose derivatives of order extend continuously to .

Weighted composition operators from weighted Bergman spaces with Békollé weights to Bloch-type spaces

Let σ be a Bekolle weight function and ν be a weight function. In this paper, we characterize the boundedness and compactness of weighted composition operators acting from Bergman-type spaces $A^{p}(\sigma)$ to Bloch-type spaces $\mathcal{B}_{\nu}$ and $\mathcal{B}_{\nu, 0}$ , considerably extending some results in the literature.

On the structure of polyharmonic Bergman-type spaces over the unit disk

Let be a pair of nonnegative integers. We deduce explicit representations for the projections onto spaces of square integrable -polyanalytic functions in terms of the two-sided compression of the Beurling–Ahlfors transform to the unit disk. We show that the space of square integrable -polyanalytic functions is a reproducing kernel Hilbert space and we deduce representations for one-to-one bounded operators from the Bergman space onto the true poly-Bergman spaces and from the harmonic Bergman space onto the true polyharmonic Bergman spaces. Moreover, we prove a decomposition theorem for polyharmonic functions in terms of their harmonic components. Finally, for positive integers we establish an isometry between a subspace of the true polyharmonic Bergman space of order with codimension and the true polyharmonic Bergman space of order .

On Traces in Some Analytic Spaces in Bounded Strictly Pseudoconvex Domains

New sharp estimates of traces of Bergman type spaces of analytic functions in bounded strictly pseudoconvex domains are obtained. These are, as far as we know, the first results of this type which are valid for any bounded strictly pseudoconvex domains with smooth boundary.

Erratum to: Planar Beurling Transform and Bergman Type Spaces

We proceed with the necessary modifications in the formulation of the second lines of (2.8), (2.9) and (2.11), (2.12) of Theorem 2.1 and Corollary 2.2, respectively, included in the original article. The second lines of (2.8) and (2.9) of Theorem 2.1 of the original article should be replaced by (1.2) and (1.4) below, respectively. For the reader’s convenience, we proceed with one proof of the later assertion.

A reproducing kernel thesis for operators on Bergman-type function spaces

In this paper we consider the reproducing kernel thesis for boundedness and compactness for various operators on Bergman-type spaces. In particular, the results in this paper apply to the weighted Bergman space on the unit ball, the unit polydisc and more generally to weighted Fock spaces.

Cyclicity in weighted Bergman type spaces

We use the resolvent transform method to study the cyclicity of the one point mass singular inner function in weighted Bergman type spaces.

Weighted Composition Operators Acting between Kind of Weighted Bergman-Type Spaces and the Bers-Type Space

In this paper, we study the boundedness and compactness of the weighted composition operator Wu,φ, which is induced by an holomorphic function u and holomorphic self-map φ, acting between the NK -space and the Bers-type space H∞ α on the unit disk. Keywords—Weighted composition operators,NK -space, Bers-type space.

On Extremal Problems in Certain New Bergman Type Spaces in Some Bounded Domains inCn

Based on recent results on boundedness of Bergman projection with positive Bergman kernel in analytic spaces in various types of domains inCn, we extend our previous sharp results on distances obtained for analytic Bergman type spaces in unit disk to some new Bergman type spaces in Lie ball, bounded symmetric domains of tube type, Siegel domains, and minimal bounded homogeneous domains.