2019/12/08 by Kruse, Karsten
#32A45 #35A01 #46A13 #46A63 #46F15 #46M20 #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.1912.03659
This work is dedicated to the development of the theory of Fourier hyperfunctions in one variable with values in a complex non-necessarily metrisable 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.