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Extension of vector-valued holomorphic and harmonic functions. (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 ${ℝ}^{N}$, $ℱ$ 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 ${f}_{u}\in ℱ\left({\Omega }\right)$ for each $u$ in a separating subset of ${E}^{\text{'}}$. Does $f$ then have an extension $F$ belonging to the space $ℱ\left({\Omega },E\right)$ of vector valued $ℱ\left({\Omega }\right)$-functions?

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

${ℱ}_{G}\left(M,E\right):=\left\{f:M\to E;\phantom{\rule{4pt}{0ex}}\forall u\in G\phantom{\rule{4pt}{0ex}}\exists {f}_{u}\in ℱ\left({\Omega }\right):{\partial }^{\alpha }{f}_{u}\left(x\right)=u\circ f\left(x,\alpha \right),\phantom{\rule{4pt}{0ex}}\left(x,\alpha \right)\in M\right\}$

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 $ℱ\left({\Omega },E\right)$) 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 $ℱ{\left({\Omega }\right)}^{\text{'}}$ 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