Vector-valued Fourier hyperfunctions and boundary values

TL;DR

This paper develops the theory of vector-valued Fourier hyperfunctions in complex locally convex spaces, establishing conditions for their existence and representing them as boundary values of holomorphic functions.

Contribution

It provides necessary and sufficient conditions for the existence of an $E$-valued Fourier hyperfunction theory, especially for ultrabornological PLS-spaces with property $(PA)$.

Findings

Characterized when $E$-valued Fourier hyperfunctions exist.

Identified spaces with and without property $(PA)$.

Represented hyperfunctions as boundary values of holomorphic functions.

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.