Extension and restriction of holomorphic functions

Diederich, Klas; Mazzilli, Emmanuel

Annales de l'Institut Fourier, Tome 47 (1997) no. 4, pp. 1079-1099.

De fortes pathologies par rapport aux propriétés de croissance peuvent apparaître pour l’extension des fonctions holomorphes d’une sous-variété $D^{\prime }$ d’un domaine $D$ à $D$ tout entier, même dans des cas très simples; les espaces $A^p(D^{\prime }):=𝒪(D^{\prime })\cap L^p(D^{\prime })$ ne sont, en général, pas tous préservés par extension. De même, la restriction de l’espace de Hilbert $A^2\left(D\right)$ à $D^{\prime }$, peut avoir une structure très étrange.

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

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