Vector-valued Fourier hyperfunctions and boundary values

Abstract

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.

References in corpus (9)

• Weighted vector-valued functions and the $\epsilon$-product
• The approximation property for weighted spaces of differentiable functions
• Parameter dependence of solutions of the Cauchy-Riemann equation on spaces of weighted smooth functions
• Vector-valued holomorphic functions in several variables
• On the nuclearity of weighted spaces of smooth functions
• The inhomogeneous Cauchy-Riemann equation for weighted smooth vector-valued functions on strips with holes
• Surjectivity of the $\overline{\partial }$-operator between weighted spaces of smooth vector-valued functions
• Series representations in spaces of vector-valued functions via Schauder decompositions
• Extension of vector-valued functions and sequence space representation