Session 17. Functional Analysis: relations to Complex Analysis and PDE

Vector-valued Fourier hyperfunctions

Karsten Kruse, Technische Universität Hamburg-Harburg, Germany
Scalar-valued Fourier hyperfunctions were introduced by Kawai [1] in 1970 as a generalization of the sheaf of hyperfunctions which was introduced by Sato [2] ([3]). He constructed them as a flabby sheaf on the radial compactification of \(\mathbb{R}^{n}\) and it turned out that the global sections are stable under Fourier transformation. Vector-valued counterparts for the theory of (Fourier) hyperfunctions were developed, at first, with values in Frèchet spaces and in 2008 Domański and Langenbruch [4] not only extended the theory of hyperfunctions far beyond the class of Frèchet spaces by using new results on splitting theory for PLS-spaces, but they also found natural limits of this kind of theory.

My talk summarizes results of my thesis [5] and is dedicated to the development of the theory of Fourier hyperfunctions in one variable with values in a non-necessarily metrizable locally convex 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 \((PA).\) It turns out 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
  1. T. Kawai, On the theory of Fourier hyperfunctions and its applications to partial differential equations with constant coefficients , J. Fac. Sci. Univ. Tokyo Sect. IA 17, 1970, 467-517.
  2. M. Sato, Theory of hyperfunctions, I , J. Fac. Sci. Univ. Tokyo Sect. I 8, 1959, 139-193.
  3. M. Sato, Theory of hyperfunctions, II , J. Fac. Sci. Univ. Tokyo Sect. I 8, 1960, 387-437.
  4. P. Domański, M. Langenbruch, Vector valued hyperfunctions and boundary values of vector valued harmonic and holomorphic functions , Publ. RIMS Kyoto Univ. 44, 2008, 1097-1142.
  5. K. Kruse, Vector-valued Fourier hyperfunctions , PhD thesis Universität Oldenburg, Oldenburg, 2014.
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