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.

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...

Certain partial differential subordinations on some Reinhardt domains in $ℂ^{n}$

Gabriela Kohr, Mirela Kohr (1997)

Annales Polonici Mathematici

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We obtain an extension of Jack-Miller-Mocanu’s Lemma for holomorphic mappings defined in some Reinhardt domains in $ℂ^{n}$ . Using this result we consider first and second order partial differential subordinations for holomorphic mappings defined on the Reinhardt domain $B_{2 p}$ with p ≥ 1.

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. ...

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...

Holomorphic series expansion of functions of Carleman type

Taib Belghiti (2004)

Annales Polonici Mathematici

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Let f be a holomorphic function of Carleman type in a bounded convex domain D of the plane. We show that f can be expanded in a series f = ∑ₙfₙ, where fₙ is a holomorphic function in Dₙ satisfying $s u p_{z \in D ₙ} | f ₙ (z) | \leq C ϱ ⁿ$ for some constants C > 0 and 0 < ϱ < 1, and where (Dₙ)ₙ is a suitably chosen sequence of decreasing neighborhoods of the closure of D. Conversely, if f admits such an expansion then f is of Carleman type. The decrease of the sequence Dₙ characterizes the smoothness of f. ...

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.

Extension from linear subvarieties for the Bergman scale of spaces on convex domains

Michał Jasiczak (2014)

Annales Polonici Mathematici

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We study the problem of extending functions from linear affine subvarieties for the Bergman scale of spaces on convex finite type domains. Our results solve the problem for H¹(D). For other Bergman spaces the result is ϵ-optimal.

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...