Kohomologie mit Schranken und Fortsetzung holomorpher Funktionen durch lineare stetige Operatoren

Abstract

In this thesis we solve the coboundary equation $\delta c=d$ with bounds for cochains with values in a coherent subsheaf of some ${\mathcal{O}}_{\Omega }^p$, 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 ${\mathcal{O}}_{\Omega }^p$, 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, $m\in \mathbb{N}$. Aytuna asked how much, and what kind of, information about the complex analytic structure of V is carried by the Fréchet 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.

104 pages