*(English)*Zbl 1141.46017

The authors present a unified treatment of the extension of holomorphic or harmonic vector valued functions, including the case of several variables. This is closely related to the investigation of conditions ensuring that a weakly holomorphic function with values in a locally convex space is holomorphic. The problem is the following: Let ${\Omega}$ be an open subset of ${\mathbb{R}}^{N}$, $\mathcal{F}$ a sheaf of smooth functions on ${\Omega}$, and let $f:M\to E$ be a function from a subset $M$ of ${\Omega}$ into a locally convex space $E$ such that $u\circ f$ has a unique extension ${f}_{u}\in \mathcal{F}\left({\Omega}\right)$ for each $u$ in a separating subset of ${E}^{\text{'}}$. Does $f$ then have an extension $F$ belonging to the space $\mathcal{F}({\Omega},E)$ of vector valued $\mathcal{F}\left({\Omega}\right)$-functions?

One of the main results here reads as follows: Let ${\Omega}$ be an open and connected subset of ${\mathbb{R}}^{N}$ and $\mathcal{F}$ a closed subsheaf of ${\mathcal{C}}^{\infty}$ over ${\Omega}$. Let $M\subset {\Omega}\times {\mathbb{N}}_{0}$ be a set of uniqueness for $\mathcal{F}\left({\Omega}\right)$ (i.e., $g\in \mathcal{F}\left({\Omega}\right)$ vanishes whenever ${\partial}^{\alpha}g\left(x\right)=0$ for all $(x,\alpha )\in M$) and let $G$ be a subspace of the dual of a locally complete space $E$ which determines boundedness (i.e., each $\sigma (E,G)$-bounded subset of $E$ is also bounded in $E$). Then the restriction map from $\mathcal{F}({\Omega},E)$ to

is surjective. The theorem extends previous theorems of Gramsch, Grosse-Erdmann and Arendt and Nikolski.

The proof uses powerful abstract techniques like the $\epsilon $-product of Laurent Schwartz (which already occurs in the definition of the vector valued sheaf $\mathcal{F}({\Omega},E)$) and results on Fréchet-Schwartz and (DFS)-spaces. A similar problem concerning the extension of locally bounded functions is also treated. An example is given which solves an extension problem of *K.-G. Grosse-Erdmann* [Math. Proc. Camb. Philos. Soc. 136, 399–411 (2004; Zbl 1055.46026)] in the negative. The last section deals with a description of the dual space $\mathcal{F}{\left({\Omega}\right)}^{\text{'}}$ of the type which *J. Wolff* [C. R. Acad. Sci. Paris 173, 1327–1328 (1921; JFM 48.0320.01)] had proved.

##### MSC:

46E40 | Spaces of vector- and operator-valued functions |

46A04 | Locally convex Fréchet spaces, etc. |

46M05 | Tensor products of topological linear spaces |