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arXiv · 1912.03659

Vector-valued Fourier hyperfunctions and boundary values

Karsten Kruse
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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.

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Karsten Kruse. 2025-11-05. Vector-valued Fourier hyperfunctions and boundary values. https://doi.org/10.1215/21562261-2024-0030

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