Kohomologie mit Schranken und Fortsetzung holomorpher Funktionen durch lineare stetige Operatoren

Details Kohomologie mit Schranken und Fortsetzung holomorpher Funktionen durch lineare stetige Operatoren Kohomologie mit Schranken und Fortsetzung holomorpher Funktionen durch lineare stetige Operatoren

2002-02-25

arxiv.org/abs/math/0202263v1

Mathematics

Functional Analysis

104 pages

Scientific paper

In this thesis we solve the coboundary equation $\delta c=d$ with bounds for cochains with values in a coherent subsheaf of some $\mathcal{O}^p_{\Omega}$, where $\Omega$ is a Stein manifold. In particular the existence of a finite set of global generators is not assumed. Our result applies therefore to the ideal sheaf $\mathcal{J}_V\subset \mathcal{O}_{\C^N}$ of germs of holomorphic functions vanishing on a closed analytic submanifold $V\subset\C^N$. Although we are mainly interested in the estimates for the solutions of $\delta c=d$, the techniques used also lead to a proof for the classical Theorem B of Cartan for coherent subsheafs of some $\mathcal{O}^p_{\Omega}$, avoiding the Mittag-Leffler argument. We derive an extension theorem for holomorphic functions on V to entire functions, with control on growth behaviour. \newline As a corollary we construct a linear tame extension operator $H(V)\to H(\C^N)$ under the hypothesis that H(V) is linear tamely isomorphic to the infinite type power series space $\Lambda_\infty(k^{\frac{1}{n}})$, n= dim$_{\C}V$; this condition is also necessary. Here the supnorms on H(V) are taken over intersections of V with polycylinders of polyradii e^m, $m\in \N$. Aytuna asked how much, and what kind of, information about the complex analytic structure of V is carried by the Fr\'echet space H(V). We prove that H(V) is linear tamely isomorphic to a power series space of infinite type if and only if V is algebraic.

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