Vector-valued Fourier hyperfunctions and boundary values

This work is dedicated to the development of the theory of Fourier hyperfunctions in one variable with values in a complex non-necessarily metrizable locally convex Hausdorff space E. Moreover, necessary and sufficient conditions are described such that a reasonable theory of E-valued Fourier hyperfunctions exists. In particular, if E is an ultrabornological PLS space, such a theory is possible if and only if E satisfies the so-called property (PA). Furthermore, many examples of such spaces having (PA) (resp. not having (PA)) are provided. We also prove that the vector-valued Fourier hyperfunctions can be realized as the sheaf generated by equivalence classes of certain compactly supported E-valued functionals and interpreted as boundary values of slowly increasing holomorphic functions.