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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 Ω be an open subset of N , a sheaf of smooth functions on Ω, and let f:ME be a function from a subset M of Ω into a locally convex space E such that uf has a unique extension f u (Ω) for each u in a separating subset of E ' . Does f then have an extension F belonging to the space (Ω,E) of vector valued (Ω)-functions?

One of the main results here reads as follows: Let Ω be an open and connected subset of N and a closed subsheaf of 𝒞 over Ω. Let MΩ× 0 be a set of uniqueness for (Ω) (i.e., g(Ω) vanishes whenever α g(x)=0 for all (x,α)M) and let G be a subspace of the dual of a locally complete space E which determines boundedness (i.e., each σ(E,G)-bounded subset of E is also bounded in E). Then the restriction map from (Ω,E) to

G (M,E):={f:ME;uGf u (Ω): α f u (x)=uf(x,α),(x,α)M}

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

The proof uses powerful abstract techniques like the ε-product of Laurent Schwartz (which already occurs in the definition of the vector valued sheaf (Ω,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 (Ω) ' of the type which J. Wolff [C. R. Acad. Sci. Paris 173, 1327–1328 (1921; JFM 48.0320.01)] had proved.


MSC:
46E40Spaces of vector- and operator-valued functions
46A04Locally convex Fréchet spaces, etc.
46M05Tensor products of topological linear spaces