On the Bergman distance on model domains in ℂⁿ

Gregor Herbort

Displaying similar documents to “On the Bergman distance on model domains in ℂⁿ”

On boundary behaviour of the Bergman projection on pseudoconvex domains

M. Jasiczak (2005)

Studia Mathematica

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It is shown that on strongly pseudoconvex domains the Bergman projection maps a space $L v_{k}$ of functions growing near the boundary like some power of the Bergman distance from a fixed point into a space of functions which can be estimated by the consecutive power of the Bergman distance. This property has a local character. Let Ω be a bounded, pseudoconvex set with C³ boundary. We show that if the Bergman projection is continuous on a space $E \supset L^{\infty} (Ω)$ defined by weighted-sup seminorms and equipped...

Weighted generalization of the Ramadanov's theorem and further considerations

Zbigniew Pasternak-Winiarski, Paweł Wójcicki (2018)

Czechoslovak Mathematical Journal

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We study the limit behavior of weighted Bergman kernels on a sequence of domains in a complex space $ℂ^{N}$ , and show that under some conditions on domains and weights, weighed Bergman kernels converge uniformly on compact sets. Then we give a weighted generalization of the theorem given by M. Skwarczyński (1980), highlighting some special property of the domains, on which the weighted Bergman kernels converge uniformly. Moreover, we show that convergence of weighted Bergman kernels implies...

The Bergman projection on weighted spaces: L¹ and Herz spaces

Oscar Blasco, Salvador Pérez-Esteva (2002)

Studia Mathematica

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We find necessary and sufficient conditions on radial weights w on the unit disc so that the Bergman type projections of Forelli-Rudin are bounded on L¹(w) and in the Herz spaces $K_{p}^{q} (w)$ .

The Bergman projection in spaces of entire functions

Jocelyn Gonessa, El Hassan Youssfi (2012)

Annales Polonici Mathematici

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We establish $L^{p}$ -estimates for the weighted Bergman projection on a nonsingular cone. We apply these results to the weighted Fock space with respect to the minimal norm in ℂⁿ.

Commutant of multiplication operators in weighted Bergman spaces on polydisk

Ali Abkar (2020)

Czechoslovak Mathematical Journal

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We study a certain operator of multiplication by monomials in the weighted Bergman space both in the unit disk of the complex plane and in the polydisk of the $n$ -dimensional complex plane. Characterization of the commutant of such operators is given.

$L ²_{h}$ -domains of holomorphy and the Bergman kernel

Peter Pflug, Włodzimierz Zwonek (2002)

Studia Mathematica

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We give a characterization of $L ²_{h}$ -domains of holomorphy with the help of the boundary behavior of the Bergman kernel and geometric properties of the boundary, respectively.

Weighted $L^{\infty}$ -estimates for Bergman projections

José Bonet, Miroslav Engliš, Jari Taskinen (2005)

Studia Mathematica

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We consider Bergman projections and some new generalizations of them on weighted $L^{\infty} ()$ -spaces. A new reproducing formula is obtained. We show the boundedness of these projections for a large family of weights v which tend to 0 at the boundary with a polynomial speed. These weights may even be nonradial. For logarithmically decreasing weights bounded projections do not exist. In this case we instead consider the projective description problem for holomorphic inductive limits.

On some extremal problems in Bergman spaces in weakly pseudoconvex domains

Romi F. Shamoyan, Olivera R. Mihić (2018)

Communications in Mathematics

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We consider and solve extremal problems in various bounded weakly pseudoconvex domains in $ℂ^{n}$ based on recent results on boundedness of Bergman projection with positive Bergman kernel in Bergman spaces $A_{α}^{p}$ in such type domains. We provide some new sharp theorems for distance function in Bergman spaces in bounded weakly pseudoconvex domains with natural additional condition on Bergman representation formula.

The Bergman kernel: Explicit formulas, deflation, Lu Qi-Keng problem and Jacobi polynomials

Tomasz Beberok (2017)

Czechoslovak Mathematical Journal

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We investigate the Bergman kernel function for the intersection of two complex ellipsoids ${(z, w_{1}, w_{2}) \in ℂ^{n + 2} : | z_{1} |^{2} + \dots + | z_{n} |^{2} + | w_{1} |^{q} < 1, | z_{1} |^{2} + \dots + | z_{n} |^{2} + | w_{2} |^{r} < 1} .$ We also compute the kernel function for ${(z_{1}, w_{1}, w_{2}) \in ℂ^{3} : | z_{1} |^{2 / n} + | w_{1} |^{q} < 1, | z_{1} |^{2 / n} + | w_{2} |^{r} < 1}$ and show deflation type identity between these two domains. Moreover in the case that $q = r = 2$ we express the Bergman kernel in terms of the Jacobi polynomials. The explicit formulas of the Bergman kernel function for these domains enables us to investigate whether the Bergman kernel has zeros or not. This kind of problem is called a Lu Qi-Keng problem. ...

Compact operators on the weighted Bergman space A¹(ψ)

Tao Yu (2006)

Studia Mathematica

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We show that a bounded linear operator S on the weighted Bergman space A¹(ψ) is compact and the predual space A₀(φ) of A¹(ψ) is invariant under S* if and only if $S k_{z} \to 0$ as z → ∂D, where $k_{z}$ is the normalized reproducing kernel of A¹(ψ). As an application, we give conditions for an operator in the Toeplitz algebra to be compact.

Weighted sub-Bergman Hilbert spaces

Maria Nowak, Renata Rososzczuk (2014)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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We consider Hilbert spaces which are counterparts of the de Branges-Rovnyak spaces in the context of the weighted Bergman spaces $A_{α}^{2}$ , $- 1 < α < \infty$ . These spaces have already been studied in [8], [7], [5] and [1]. We extend some results from these papers.

Two-sided $L^{1}$ -estimates for Szegö kernels on classical domains

Josephine Mitchell, G. Sampson (1990)

Annales Polonici Mathematici

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On the Green function on a certain class of hyperconvex domains

Gregor Herbort (2008)

Annales Polonici Mathematici

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We study the behavior of the pluricomplex Green function on a bounded hyperconvex domain D that admits a smooth plurisubharmonic exhaustion function ψ such that 1/|ψ| is integrable near the boundary of D, and moreover satisfies the estimate $| ψ | \leq C e x p (- C^{'} {(l o g (1 / δ_{D}))}^{α})$ at points close enough to the boundary with constants C,C’ > 0 and 0 < α < 1. Furthermore, we obtain a Hopf lemma for such a function ψ. Finally, we prove a lower bound on the Bergman distance on D.

Inclusion relations between harmonic Bergman-Besov and weighted Bloch spaces on the unit ball

Ömer Faruk Doğan, Adem Ersin Üreyen (2019)

Czechoslovak Mathematical Journal

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We consider harmonic Bergman-Besov spaces $b_{α}^{p}$ and weighted Bloch spaces $b_{α}^{\infty}$ on the unit ball of $ℝ^{n}$ for the full ranges of parameters $0 < p < \infty$ , $α \in ℝ$ , and determine the precise inclusion relations among them. To verify these relations we use Carleson measures and suitable radial differential operators. For harmonic Bergman spaces various characterizations of Carleson measures are known. For weighted Bloch spaces we provide a characterization when $α > 0$ .

Compactness of composition operators acting on weighted Bergman-Orlicz spaces

Ajay K. Sharma, S. Ueki (2012)

Annales Polonici Mathematici

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We characterize compact composition operators acting on weighted Bergman-Orlicz spaces $_{α}^{ψ} = f \in H () : \int_{} ψ (| f (z) |) d A_{α} (z) < \infty$ , where α > -1 and ψ is a strictly increasing, subadditive convex function defined on [0,∞) and satisfying ψ(0) = 0, the growth condition $l i m_{t \to \infty} ψ (t) / t = \infty$ and the Δ₂-condition. In fact, we prove that $C_{φ}$ is compact on $_{α}^{ψ}$ if and only if it is compact on the weighted Bergman space $²_{α}$ .

On locally convex extension of $H^{\infty}$ in the unit ball and continuity of the Bergman projection

M. Jasiczak (2003)

Studia Mathematica

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We define locally convex spaces LW and HW consisting of measurable and holomorphic functions in the unit ball, respectively, with the topology given by a family of weighted-sup seminorms. We prove that the Bergman projection is a continuous map from LW onto HW. These are the smallest spaces having this property. We investigate the topological and algebraic properties of HW.

The pluricomplex Green function on some regular pseudoconvex domains

Gregor Herbort (2014)

Annales Polonici Mathematici

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Let D be a smooth bounded pseudoconvex domain in ℂⁿ of finite type. We prove an estimate on the pluricomplex Green function $_{D} (z, w)$ of D that gives quantitative information on how fast the Green function vanishes if the pole w approaches the boundary. Also the Hölder continuity of the Green function is discussed.

Weighted composition operators from Zygmund spaces to Bloch spaces on the unit ball

Yu-Xia Liang, Chang-Jin Wang, Ze-Hua Zhou (2015)

Annales Polonici Mathematici

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Let H() denote the space of all holomorphic functions on the unit ball ⊂ ℂⁿ. Let φ be a holomorphic self-map of and u∈ H(). The weighted composition operator $u C_{φ}$ on H() is defined by $u C_{φ} f (z) = u (z) f (φ (z))$ . We investigate the boundedness and compactness of $u C_{φ}$ induced by u and φ acting from Zygmund spaces to Bloch (or little Bloch) spaces in the unit ball.

The Properties of the Weighted Space $H_{2, α}^{k} (Ω)$ and Weighted Set $W_{2, α}^{k} (Ω, δ)$

V.A. Rukavishnikov, E.V. Matveeva, E.I. Rukavishnikova (2018)

Communications in Mathematics

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We study the properties of the weighted space $H_{2, α}^{k} (Ω)$ and weighted set $W_{2, α}^{k} (Ω, δ)$ for boundary value problem with singularity.

Existence and regularity of solutions of some elliptic system in domains with edges

Wojciech M. Zajączkowski

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CONTENTS1. Introduction.......................................................................52. Notation and auxiliary results............................................93. Statement of the problem (1.1)-(1.3)..............................204. The problem (3.14).........................................................225. Auxiliary results in $D_{ϑ}$ ...............................................346. Existence of solutions of (3.14) in $H_{μ}^{k} (D_{ϑ})$ ............417. Green function................................................................528....

On some new sharp embedding theorems in minimal and pseudoconvex domains

Romi F. Shamoyan, Olivera R. Mihić (2016)

Czechoslovak Mathematical Journal

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We present new sharp embedding theorems for mixed-norm analytic spaces in pseudoconvex domains with smooth boundary. New related sharp results in minimal bounded homogeneous domains in higher dimension are also provided. Last domains we consider are domains which are direct generalizations of the well-studied so-called bounded symmetric domains in $ℂ^{n} .$ Our results were known before only in the very particular case of domains of such type in the unit ball. As in the unit ball case, all our...

Disjoint hypercyclic powers of weighted translations on groups

Liang Zhang, Hui-Qiang Lu, Xiao-Mei Fu, Ze-Hua Zhou (2017)

Czechoslovak Mathematical Journal

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Let $G$ be a locally compact group and let $1 \leq p < \infty .$ Recently, Chen et al. characterized hypercyclic, supercyclic and chaotic weighted translations on locally compact groups and their homogeneous spaces. There has been an increasing interest in studying the disjoint hypercyclicity acting on various spaces of holomorphic functions. In this note, we will study disjoint hypercyclic and disjoint supercyclic powers of weighted translation operators on the Lebesgue space $L^{p} (G)$ in terms of the weights. Sufficient...

Hilbert-Schmidt Hankel operators with anti-holomorphic symbols on complete pseudoconvex Reinhardt domains

Mehmet Çelik, Yunus E. Zeytuncu (2017)

Czechoslovak Mathematical Journal

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On complete pseudoconvex Reinhardt domains in $ℂ^{2}$ , we show that there is no nonzero Hankel operator with anti-holomorphic symbol that is Hilbert-Schmidt. In the proof, we explicitly use the pseudoconvexity property of the domain. We also present two examples of unbounded non-pseudoconvex domains in $ℂ^{2}$ that admit nonzero Hilbert-Schmidt Hankel operators with anti-holomorphic symbols. In the first example the Bergman space is finite dimensional. However, in the second example the Bergman...

Operator positivity and analytic models of commuting tuples of operators

Monojit Bhattacharjee, Jaydeb Sarkar (2016)

Studia Mathematica

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We study analytic models of operators of class $C_{\cdot 0}$ with natural positivity assumptions. In particular, we prove that for an m-hypercontraction $T \in C_{\cdot 0}$ on a Hilbert space , there exist Hilbert spaces and ⁎ and a partially isometric multiplier θ ∈ ℳ (H²(),A²ₘ(⁎)) such that $≅_{θ} = A ² ₘ (⁎) ⊖ θ H ² ()$ and $T ≅ P_{_{θ}} M_{z} |_{_{θ}}$ , where A²ₘ(⁎) is the ⁎-valued weighted Bergman space and H²() is the -valued Hardy space over the unit disc . We then proceed to study analytic models for doubly commuting n-tuples of operators and investigate their...

Existence of solutions to the (rot,div)-system in $L_{p}$ -weighted spaces

Wojciech M. Zajączkowski (2010)

Applicationes Mathematicae

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The existence of solutions to the elliptic problem rot v = w, div v = 0 in a bounded domain Ω ⊂ ℝ³, ${v \cdot n ̅ |}_{S} = 0$ , S = ∂Ω in weighted $L_{p}$ -Sobolev spaces is proved. It is assumed that an axis L crosses Ω and the weight is a negative power function of the distance to the axis. The main part of the proof is devoted to examining solutions of the problem in a neighbourhood of L. The existence in Ω follows from the technique of regularization.