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Extension and restriction of holomorphic functions

Klas Diederich;  Emmanuel Mazzilli — 1997

Annales de l'institut Fourier

Strong pathologies with respect to growth properties can occur for the extension of holomorphic functions from submanifolds D ' of pseudoconvex domains D to all of D even in quite simple situations; The spaces A p ( D ' ) : = 𝒪 ( D ' ) ∩ L p ( D ' ) are, in general, not at all preserved. Also the image of the Hilbert space A 2 ( D ) under the restriction to D ' can have a very strange structure.

Quantitative estimates for the Green function and an application to the Bergman metric

Klas Diederich;  Gregor Herbort — 2000

Annales de l'institut Fourier

Let D ⊂ ℂ n be a bounded pseudoconvex domain that admits a Hölder continuous plurisubharmonic exhaustion function. Let its pluricomplex Green function be denoted by G D ( . , . ) . In this article we give for a compact subset K ⊂ D a quantitative upper bound for the supremum sup z ∈ K | G D ( z , w ) | in terms of the boundary distance of K and w . This enables us to prove that, on a smooth bounded regular domain D (in the sense of Diederich-Fornaess), the Bergman differential metric B D ( w ; X ) tends to infinity, for X ∈ ℂ n / { O } , when w ∈ D tends to a boundary point....

Real and complex analytic sets. The relevance of Segre varieties

Klas Diederich;  Emmanuel Mazzilli — 2008

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Let X ⊂ ℂ n n be a closed real-analytic subset and put 𝒜 : = { z ∈ X ∣ ∃ A ⊂ X , germ of a complex-analytic set, z ∈ A , dim z A > 0 } This article deals with the question of the structure of 𝒜 . In the main result a natural proof is given for the fact, that 𝒜 always is closed. As a main tool an interesting relation between complex analytic subsets of X of positive dimension and the Segre varieties of X is proved and exploited.

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