We develop the theory of hyperfunctions with values in a locally convex nonnecessarily metrizable space E and ﬁnd necessary conditions and sufficient conditions such that a reasonable theory of E-valued hyperfunctions exists. In particular, we show that it exists for various spaces of distributions but there is no such theory for the spaces of real analytic functions and distributions with compact support. We also show that vector valued hyperfunctions can be interpreted as boundary values of vector valued harmonic or holomorphic functions and, in many cases, as suitable cohomology groups.
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Paweł Domański, Michael Langenbruch, Vector Valued Hyperfunctions and Boundary Values of Vector Valued Harmonic and Holomorphic Functions. Publ. Res. Inst. Math. Sci. 44 (2008), no. 4, pp. 1097–1142DOI 10.2977/PRIMS/1231263781