Vector-valued Fourier hyperfunctions and boundary values
Karsten Kruse
TL;DR
This work develops a theory of one-variable Fourier hyperfunctions valued in complex locally convex spaces $E$, including non-metrisable cases, by leveraging Silva–Köthe–Grothendieck duality and a duality-method construction. It shows that a workable $E$-valued theory exists for ultrabornological PLS-spaces exactly when $E$ has property $(PA)$, and it realizes these hyperfunctions as a flabby sheaf generated by compactly supported $E$-valued $ ext{P}_{*}$-functionals, while also interpreting them as boundary values of slowly increasing holomorphic $E$-valued functions. A key contribution is the boundary-value representation of $L_b(oldsymbol{ ext{P}}_{*}(ar{oldsymbol{ extR}}),E)$ and the associated topological isomorphisms, enabling a well-defined Fourier transform $oldsymbol{ extsf{F}}_{oldsymbol{ extstar}}$ that preserves the vector-valued structure. The paper also establishes that strict admissibility and the $(PA)$ property are pivotal to the theory’s existence, mapping out a landscape of affirmative and negative examples and addressing open questions from prior vector-valued hyperfunction work.
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.