On weights which admit the reproducing kernel of Bergman type

Pasternak-Winiarski, Zbigniew

doi:10.1155/s0161171292000012

Cited by 22 publications

(19 citation statements)

References 8 publications

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“…A weight γe W(Ω) is admissible (cf. [17]) if 1) L 2 H(Ω,γ) is a closed subspace of L 2 (Ω,y), and 2) for any zeΩ the evaluation functional δ z :…”

Section: The Forelli-rudin-ligocka-peloso Asymptotic Expansion Formulamentioning

confidence: 99%

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On boundary behaviour of symplectomorphisms

Barletta¹,

Dragomir²

1998

Kodai Math. J.

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Let Ω c C" be a strictly pseudoconvex domain, γ an admissible weight, and K γ (z, C) the reproducing (or y-Bergman) kernel for L 2 //(Ω,y), the space of square integrable functions, with respect to the measure γdμ, which are holomorphic in Ω (dμ is the Lebesgue measure in R 2n ), cf. e.g. Z. . Consider the complex tensor field: and the corresponding real tangent (0,2)-tensor field g γ given by:where χ(Ω) is the C°°(Ω)-module of all real tangent vector fields on Ω. Under suitable conditions (cf. section 2) g γ is a Kahlerian metric on Ω, hence ω γ = -idδ logK γ (z,z) is a symplectic structure (the Kahler 2-form of g γ ). One of the problems we take up in the present paper may be stated as follows. Let F : Ω -• Ω be a symplectomorphism of (Ω,ω y ) in itself, smooth up to the boundary. Does F : 3Ω -> 3Ω preserve the contact structure of the boundary?Our interest may be motivated as follows. If F : Ω -> Ω is a biholomorphism then, by a celebrated result of C. Fefferman (cf. Theorem 1 in [4], p. 2) F is smooth up to the boundary, hence F : 3Ω -> 3Ω is a CR diffeomorphism, and in particular a contact transformation. Also biholomorphisms are known to be isometries of the Bergman metric g\ (cf. e.g. [7], p. 370) hence symplectomorphisms of (Ω,ωi). On the other hand, one may weaken the assumption on F by requesting only that F be a C 00 diffeomorphism and F*a>\ =ω\.Then, by a result of A. Koranyi and H. M. Reimann [11], if F is smooth up to the boundary then F : 3Ω -• dΩ is a contact transformation.The main ingredient in the proof of A. Koranyi and H. M. Reimann's result is the fact that, when γ = 1, a certain negative power of the Bergman kernel (p(z) = K\(z,z)~ι^n +i>} ) is a defining function of Ω (allowing one to relate the symplectic structure of Ω to the contact structure of its boundary). In turn, this is a consequence of C. Fefferman's asymptotic expansion of K\(z, ζ) (cf. Theorem 2 in

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“…A weight γe W(Ω) is admissible (cf. [17]) if 1) L 2 H(Ω,γ) is a closed subspace of L 2 (Ω,y), and 2) for any zeΩ the evaluation functional δ z :…”

Section: The Forelli-rudin-ligocka-peloso Asymptotic Expansion Formulamentioning

confidence: 99%

“…Let Ω = {φ < 0} be a domain and 7 e ^ W(Ω) and admissible weight. By a result in [17] one has the representation for any complete orthonormal system {φ k } in L 2 H(Ω,γ).…”

Section: Symplectomorphisms Of Y-kobayashi Domainsmentioning

confidence: 99%

On boundary behaviour of symplectomorphisms

Barletta¹,

Dragomir²

1998

Kodai Math. J.

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“…It is clear that £"• is a linear functional. Using similar methods as in the case of an ordinary Bergman space (see [3] or [9]) we prove the following Putting it and (9) into (10) we obtain…”

Section: {S | T) = (S \T) Ktll := F H(st)fx Merg°(e)mentioning

confidence: 86%

“…If E is a positive line bundle over a compact complex manifold M then Z is an embeding and E is induced by Z and the bundle dual to the universal bundle over the projective space CP(L 2 H(Ei)) (for the application of the maps J and Z in the proof of the Kodaira theorem see [10]). Also in the non-compact case, if the space 0(M, E) of all holomorphic sections of E is sufficiently rich and the volume form /x is appropriately chosen the Bergman section contains the whole information about E, h, and (i (see [9] [10]).…”

Section: H{e)) ¿S the R-th Grassmann Space Over L 2 H(e) •mentioning

confidence: 99%

Bergman Spaces and Kernels for Holomorphic Vector Bundles

Pasternak-Winiarski¹,

Wojcieszynski²

1997

Demonstratio Mathematica

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“…The definition of admissible weight provides us with existence and uniqueness of the related Bergman kernel and completeness of the space L 2 H (W, µ). The concept of a-weight was introduced in [13], and in [14] several theorems concerning admissible weights are proved. An illustrative result is:…”

Section: Definitions and Notationmentioning

confidence: 99%