Proper holomorphic liftings and new formulas for the Bergman and Szegő kernels

E. H. Youssfi

Displaying similar documents to “Proper holomorphic liftings and new formulas for the Bergman and Szegő kernels”

Bergman function, Genchev transform and L²-angles, for multidimensional tubes

Hyb Wojciech

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CONTENTS1. Introduction.......................................................................................................52. Basic definitions, notations and facts................................................................63. Definitions of the Genchev transform................................................................84. Basic properties of the Genchev transform......................................................115. Some properties of the weight $w_{B}$ ..............................................................166....

Some properties of Reinhardt domains

Le Mau Hai, Nguyen Quang Dieu, Nguyen Huu Tuyen (2003)

Annales Polonici Mathematici

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We first establish the equivalence between hyperconvexity of a fat bounded Reinhardt domain and the existence of a Stein neighbourhood basis of its closure. Next, we give a necessary and sufficient condition on a bounded Reinhardt domain D so that every holomorphic mapping from the punctured disk $Δ_{*}$ into D can be extended holomorphically to a map from Δ into D.

Peak functions on convex domains

Kolář, Martin

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Let $Ω \subset ℂ^{n}$ be a domain with smooth boundary and $p \in \partial Ω$ . A holomorphic function $f$ on $Ω$ is called a $C^{k}$ ( $k = 0, 1, 2, \dots$ ) peak function at $p$ if $f \in C^{k} (\bar{Ω})$ , $f (p) = 1$ , and $| f (q) | < 1$ for all $q \in \bar{Ω} ∖ {p}$ . If $Ω$ is strongly pseudoconvex, then $C^{\infty}$ peak functions exist. On the other hand, J. E. Fornaess constructed an example in $ℂ^{2}$ to show that this result fails, even for $C^{1}$ functions, on a weakly pseudoconvex domain [Math. Ann. 227, 173-175 (1977; Zbl 0346.32026)]. Subsequently, E. Bedford and J. E. Fornaess showed that there is always a continuous peak function...

On the Djrbashian kernel of a Siegel domain

Elisabetta Barletta, Sorin Dragomir (1998)

Studia Mathematica

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We establish an inversion formula for the M. M. Djrbashian A. H. Karapetyan integral transform (cf. [6]) on the Siegel domain $Ω_{n} = ζ \in ℂ^{n} : ϱ (ζ) > 0$ , $ϱ (ζ) = I m (ζ_{1}) - {| ζ^{'} |}^{2}$ . We build a family of Kähler metrics of constant holomorphic curvature whose potentials are the $ϱ^{α}$ -Bergman kernels, α > -1, (in the sense of Z. Pasternak-Winiarski [20] of $Ω_{n}$ . We build an anti-holomorphic embedding of $Ω_{n}$ in the complex projective Hilbert space $ℂ ℙ (H_{α}^{2} (Ω_{n}))$ and study (in connection with work by A. Odzijewicz [18] the corresponding transition probability...

$L ²_{h}$ -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.

Zeros of bounded holomorphic functions in strictly pseudoconvex domains in $ℂ^{2}$

Jim Arlebrink (1993)

Annales de l'institut Fourier

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Let $D$ be a bounded strictly pseudoconvex domain in $ℂ^{2}$ and let $X$ be a positive divisor of $D$ with finite area. We prove that there exists a bounded holomorphic function $f$ such that $X$ is the zero set of $f$ . This result has previously been obtained by Berndtsson in the case where $D$ is the unit ball in $ℂ^{2}$ .

A boundary cross theorem for separately holomorphic functions

Peter Pflug, Viêt-Anh Nguyên (2004)

Annales Polonici Mathematici

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Let D ⊂ ℂⁿ and $G \subset ℂ^{m}$ be pseudoconvex domains, let A (resp. B) be an open subset of the boundary ∂D (resp. ∂G) and let X be the 2-fold cross ((D∪A)×B)∪(A×(B∪G)). Suppose in addition that the domain D (resp. G) is locally ² smooth on A (resp. B). We shall determine the “envelope of holomorphy” X̂ of X in the sense that any function continuous on X and separately holomorphic on (A×G)∪(D×B) extends to a function continuous on X̂ and holomorphic on the interior of X̂. A generalization of this...

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...

Boundary behaviour of holomorphic functions in Hardy-Sobolev spaces on convex domains in ℂⁿ

Marco M. Peloso, Hercule Valencourt (2010)

Colloquium Mathematicae

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We study the boundary behaviour of holomorphic functions in the Hardy-Sobolev spaces $ℋ^{p, k} ()$ , where is a smooth, bounded convex domain of finite type in ℂⁿ, by describing the approach regions for such functions. In particular, we extend a phenomenon first discovered by Nagel-Rudin and Shapiro in the case of the unit disk, and later extended by Sueiro to the case of strongly pseudoconvex domains.

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.

An example of a pseudoconvex domain whose holomorphic sectional curvature of the Bergman metric is unbounded

Gregor Herbort (2007)

Annales Polonici Mathematici

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Let a and m be positive integers such that 2a < m. We show that in the domain $D : = {z \in ℂ ³ | r (z) : = ℜ z ₁ + | z ₁ | ² + | z ₂ |}^{2 m} + {| z ₂ z ₃ |}^{2 a} + {| z ₃ |}^{2 m} < 0$ the holomorphic sectional curvature $R_{D} (z; X)$ of the Bergman metric at z in direction X tends to -∞ when z tends to 0 non-tangentially, and the direction X is suitably chosen. It seems that an example with this feature has not been known so far.

The $\bar{\partial}$ -Neumann operator and commutators of the Bergman projection and multiplication operators

Friedrich Haslinger (2008)

Czechoslovak Mathematical Journal

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We prove that compactness of the canonical solution operator to $\bar{\partial}$ restricted to $(0, 1)$ -forms with holomorphic coefficients is equivalent to compactness of the commutator $[𝒫, \bar{M}]$ defined on the whole $L_{(0, 1)}^{2} (Ω),$ where $\bar{M}$ is the multiplication by $\bar{z}$ and $𝒫$ is the orthogonal projection of $L_{(0, 1)}^{2} (Ω)$ to the subspace of $(0, 1)$ forms with holomorphic coefficients. Further we derive a formula for the $\bar{\partial}$ -Neumann operator restricted to $(0, 1)$ forms with holomorphic coefficients expressed by commutators of the Bergman projection and the...