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

Tomasz Beberok

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Displaying similar documents to “The Bergman kernel: Explicit formulas, deflation, Lu Qi-Keng problem and Jacobi polynomials”

$L ²_{h}$ -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.

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

Josephine Mitchell, G. Sampson (1990)

Annales Polonici Mathematici

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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...

Strict plurisubharmonicity of Bergman kernels on generalized annuli

Yanyan Wang (2014)

Annales Polonici Mathematici

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Let $A_{ζ} = Ω - \bar{ρ (ζ) \cdot Ω}$ be a family of generalized annuli over a domain U. We show that the logarithm of the Bergman kernel $K_{ζ} (z)$ of $A_{ζ}$ is plurisubharmonic provided ρ ∈ PSH(U). It is remarkable that $A_{ζ}$ is non-pseudoconvex when the dimension of $A_{ζ}$ is larger than one. For standard annuli in ℂ, we obtain an interesting formula for $\partial ² l o g K_{ζ} / \partial ζ \partial ζ ̅$ , as well as its boundary behavior.

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 $K_{p}^{q} (w)$ .

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.

Espace de Dixmier des opérateurs de Hankel sur les espaces de Bergman à poids

Romaric Tytgat (2015)

Czechoslovak Mathematical Journal

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Nous donnons des résultats théoriques sur l’idéal de Macaev et la trace de Dixmier. Ensuite, nous caractérisons les symboles antiholomorphes $\bar{f}$ tels que l’opérateur de Hankel $H_{\bar{f}}$ sur l’espace de Bergman à poids soit dans l’idéal de Macaev et nous donnons la trace de Dixmier. Pour cela, nous regardons le comportement des normes de Schatten $𝒮^{p}$ quand $p$ tend vers $1$ et nous nous appuyons sur le résultat de Engliš et Rochberg sur l’espace de Bergman. Nous parlons aussi des puissances de tels opérateurs....

Zeros of functions in $D^{p, α}$ spaces

M. Jevtić (1981)

Matematički Vesnik

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Some methods of constructing kernels in statistical learning

Tomasz Górecki, Maciej Łuczak (2010)

Discussiones Mathematicae Probability and Statistics

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This paper is a collection of numerous methods and results concerning a design of kernel functions. It gives a short overview of methods of building kernels in metric spaces, especially $R^{n}$ and $S^{n}$ . However we also present a new theory. Introducing kernels was motivated by searching for non-linear patterns by using linear functions in a feature space created using a non-linear feature map.

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 $α > 0$ .

A generalization of Bateman's expansion and finite integrals of Sonine's and Feldheim's type

Giacomo Gigante (2010)

Colloquium Mathematicae

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Let ${A_{k}}_{k = 0}^{+ \infty}$ be a sequence of arbitrary complex numbers, let α,β > -1, let Pₙα,βn=0+∞ $b e t h e J a c o b i p o l y n o m i a l s a n d d e f i n e t h e f u n c t i o n s$ $H ₙ (α, z) = \sum_{m = n}^{+ \infty} (A_{m} z^{m}) / (Γ (α + n + m + 1) (m - n)!)$ , $G (α, β, x, y) = \sum_{r, s = 0}^{+ \infty} (A_{r + s} x^{r} y^{s}) / (Γ (α + r + 1) Γ (β + s + 1) r! s!)$ . Then, for any non-negative integer n, $\int_{0}^{π / 2} G (α, β, x ² s i n ² ϕ, y ² c o s ² ϕ) P ₙ^{α, β} (c o s ² ϕ) s i n^{2 α + 1} ϕ c o s^{2 β + 1} ϕ d = 1 / 2 H ₙ (α + β + 1, x ² + y ²) P ₙ^{α, β} ((y ² - x ²) / (y ² + x ²))$ . When $A_{k} = {(- 1 / 4)}^{k}$ , this formula reduces to Bateman’s expansion for Bessel functions. For particular values of y and n one obtains generalizations of several formulas already known for Bessel functions, like Sonine’s first and second finite integrals and certain Neumann series expansions. Particular choices of ${A_{k}}_{k = 0}^{+ \infty}$ allow one to write all these type of formulas...

The ideal boundary of a domain in $C^{n}$

Maciej Skwarczyński (1981)

Annales Polonici Mathematici

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Toeplitz operators on Bergman spaces and Hardy multipliers

Wolfgang Lusky, Jari Taskinen (2011)

Studia Mathematica

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We study Toeplitz operators $T_{a}$ with radial symbols in weighted Bergman spaces $A_{μ}^{p}$ , 1 < p < ∞, on the disc. Using a decomposition of $A_{μ}^{p}$ into finite-dimensional subspaces the operator $T_{a}$ can be considered as a coefficient multiplier. This leads to new results on boundedness of $T_{a}$ and also shows a connection with Hardy space multipliers. Using another method we also prove a necessary and sufficient condition for the boundedness of $T_{a}$ for a satisfying an assumption on the positivity of certain...

Extension of germs of holomorphic isometries up to normalizing constants with respect to the Bergman metric

Ngaiming Mok (2012)

Journal of the European Mathematical Society

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We study the extension problem for germs of holomorphic isometries $f : (D; x_{0}) \to (Ω; f (x_{0}))$ up to normalizing constants between bounded domains in Euclidean spaces equipped with Bergman metrics $d s_{D}^{2}$ on $D$ and $d s_{Ω}^{2}$ on $Ω$ . Our main focus is on boundary extension for pairs of bounded domains $(D, Ω)$ such that the Bergman kernel $K_{D} (z, w)$ extends meromorphically in $(z, \bar{w})$ to a neighborhood of $\bar{D} \times D$ , and such that the analogous statement holds true for the Bergman kernel $K_{Ω} (ς, ξ)$ on $Ω$ . Assuming that $(D; d s_{D}^{2})$ and $(Ω; d s_{Ω}^{2})$ are complete Kähler manifolds, we prove that...

The punctured plane: alternating projections and $L^{2}$ -angles

M. Skwarczyński (1991)

Annales Polonici Mathematici

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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 $²_{α}$ .

Jacobi decomposition of weighted Triebel-Lizorkin and Besov spaces

George Kyriazis, Pencho Petrushev, Yuan Xu (2008)

Studia Mathematica

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The Littlewood-Paley theory is extended to weighted spaces of distributions on [-1,1] with Jacobi weights $w (t) = {(1 - t)}^{α} {(1 + t)}^{β}$ . Almost exponentially localized polynomial elements (needlets) $φ_{ξ}$ , $ψ_{ξ}$ are constructed and, in complete analogy with the classical case on ℝⁿ, it is shown that weighted Triebel-Lizorkin and Besov spaces can be characterized by the size of the needlet coefficients $⟨ f, φ_{ξ} ⟩$ in respective sequence spaces.

Bounded evaluation operators from $H^{p}$ into $ℓ^{q}$

Martin Smith (2007)

Studia Mathematica

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Given 0 < p,q < ∞ and any sequence z = zₙ in the unit disc , we define an operator from functions on to sequences by $T_{z, p} (f) = {(1 - | z ₙ | ²)}^{1 / p} f (z ₙ)$ . Necessary and sufficient conditions on zₙ are given such that $T_{z, p}$ maps the Hardy space $H^{p}$ boundedly into the sequence space $ℓ^{q}$ . A corresponding result for Bergman spaces is also stated.

Schatten class generalized Toeplitz operators on the Bergman space

Chunxu Xu, Tao Yu (2021)

Czechoslovak Mathematical Journal

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Let $μ$ be a finite positive measure on the unit disk and let $j \geq 1$ be an integer. D. Suárez (2015) gave some conditions for a generalized Toeplitz operator $T_{μ}^{(j)}$ to be bounded or compact. We first give a necessary and sufficient condition for $T_{μ}^{(j)}$ to be in the Schatten $p$ -class for $1 \leq p < \infty$ on the Bergman space $A^{2}$ , and then give a sufficient condition for $T_{μ}^{(j)}$ to be in the Schatten $p$ -class $(0 < p < 1)$ on $A^{2}$ . We also discuss the generalized Toeplitz operators with general bounded symbols. If $ϕ \in L^{\infty} (D, d A)$ and $1 < p < \infty$ , we define the generalized...

A Hankel matrix acting on Hardy and Bergman spaces

Petros Galanopoulos, José Ángel Peláez (2010)

Studia Mathematica

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Let μ be a finite positive Borel measure on [0,1). Let $ℋ_{μ} = {(μ_{n, k})}_{n, k \geq 0}$ be the Hankel matrix with entries $μ_{n, k} = \int_{[0, 1)} t^{n + k} d μ (t)$ . The matrix $_{μ}$ induces formally an operator on the space of all analytic functions in the unit disc by the fomula $ℋ_{μ} (f) (z) = \sum_{n = 0}^{\infty} i (\sum_{k = 0}^{\infty} μ_{n, k} a_{k}) z ⁿ$ , z ∈ , where $f (z) = \sum_{n = 0}^{\infty} a ₙ z ⁿ$ is an analytic function in . We characterize those positive Borel measures on [0,1) such that $ℋ_{μ} (f) (z) = \int_{[0, 1)} f (t) / (1 - t z) d μ (t)$ for all f in the Hardy space H¹, and among them we describe those for which $ℋ_{μ}$ is bounded and compact on H¹. We also study the analogous problem for the Bergman space A². ...

Complex symmetry of Toeplitz operators on the weighted Bergman spaces

Xiao-He Hu (2022)

Czechoslovak Mathematical Journal

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We give a concrete description of complex symmetric monomial Toeplitz operators $T_{z^{p} {\bar{z}}^{q}}$ on the weighted Bergman space $A^{2} (Ω)$ , where $Ω$ denotes the unit ball or the unit polydisk. We provide a necessary condition for $T_{z^{p} {\bar{z}}^{q}}$ to be complex symmetric. When $p, q \in ℕ^{2}$ , we prove that $T_{z^{p} {\bar{z}}^{q}}$ is complex symmetric on $A^{2} (Ω)$ if and only if $p_{1} = q_{2}$ and $p_{2} = q_{1}$ . Moreover, we completely characterize when monomial Toeplitz operators $T_{z^{p} {\bar{z}}^{q}}$ on $A^{2} (𝔻_{n})$ are $J_{U}$ -symmetric with the $n \times n$ symmetric unitary matrix $U$ .