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 be an open subset of , a sheaf of smooth functions on , and let be a function from a subset of into a locally convex space such that has a unique extension for each in a separating subset of . Does then have an extension belonging to the space of vector valued -functions?
One of the main results here reads as follows: Let be an open and connected subset of and a closed subsheaf of over . Let be a set of uniqueness for (i.e., vanishes whenever for all ) and let be a subspace of the dual of a locally complete space which determines boundedness (i.e., each -bounded subset of is also bounded in ). Then the restriction map from to
is surjective. The theorem extends previous theorems of Gramsch, Grosse-Erdmann and Arendt and Nikolski.
The proof uses powerful abstract techniques like the -product of Laurent Schwartz (which already occurs in the definition of the vector valued sheaf ) 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 of the type which J. Wolff [C. R. Acad. Sci. Paris 173, 1327–1328 (1921; JFM 48.0320.01)] had proved.