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

Hyb Wojciech

Displaying similar documents to “Bergman function, Genchev transform and L²-angles, for multidimensional tubes”

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

E. H. Youssfi (2002)

Studia Mathematica

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We consider a large class of convex circular domains in $M_{m ₁, n ₁} (ℂ) \times . . . \times M_{m_{d}, n_{d}} (ℂ)$ which contains the oval domains and minimal balls. We compute their Bergman and Szegő kernels. Our approach relies on the analysis of some proper holomorphic liftings of our domains to some suitable manifolds.

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...

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...

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...

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.

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 $ℂ^{2}$ .

Weighted Bergman projections and tangential area integrals

William Cohn (1993)

Studia Mathematica

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Let Ω be a bounded strictly pseudoconvex domain in $ℂ^{n}$ . In this paper we find sufficient conditions on a function f defined on Ω in order that the weighted Bergman projection $P_{s} f$ belong to the Hardy-Sobolev space $H_{k}^{p} (\partial Ω)$ . The conditions on f we consider are formulated in terms of tent spaces and complex tangential vector fields. If f is holomorphic then these conditions are necessary and sufficient in order that f belong to the Hardy-Sobolev space $H_{k}^{p} (\partial Ω)$ .

Besov spaces on bounded symmetric domains.

Jevtić, Miroljub (1997)

Matematichki Vesnik

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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...

Regularity of the Bergman Projection and Duality of Holomorphic Function Spaces.

Harold P. Boas, StevenR. Bell (1984)

Mathematische Annalen

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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...

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 Bergman kernel functions of certain unbounded domains

Friedrich Haslinger (1998)

Annales Polonici Mathematici

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We compute the Bergman kernel functions of the unbounded domains $Ω_{p} = (z^{'}, z) \in ℂ ² : z > p (z^{'})$ , where $p (z^{'}) = {| z^{'} |}^{α} / α$ . It is also shown that these kernel functions have no zeros in $Ω_{p}$ . We use a method from harmonic analysis to reduce the computation of the 2-dimensional case to the problem of finding the kernel function of a weighted space of entire functions in one complex variable.