Vector valued Fourier hyperfunctions (Q1202850)

The author studies vector valued Fourier hyperfunctions valued in a complex Hilbert space \(H\) which is not necessarily separable. This is a direct generalization of the author and \textit{Sh. Nagamachi} [Proc. Japan Acad. 51, 558-561 (1975; Zbl 0333.46031) and J. Math., Tokushima Univ. 9, 1-33 (1975; Zbl 0315.46039)]. He realizes \(H\)-valued Fourier hyperfunctions by the duality method using elements of the dual space of the space of all rapidly decreasing \(H\)-valued real analytic functions or by the algebro-analytic method as ``boundary values'' of slowly increasing \(H\)-valued holomorphic functions and then shows that they are the twofold realization of the same \(H\)-valued Fourier hyperfunctions. He proves the equivalence and mutual independence of these two methods. In case of duality method, it is characteristic that test functions are also vector valued. This idea is also used by \textit{E. Brüning} and \textit{Sh. Nagamachi} [J. Math. Phys. 30, No. 10, 2340-2359 (1989; Zbl 0703.46050)] in the somewhat different context. For that purpose, he uses the vector valued version of the method of \(L_ 2\) estimates for the \(\overline\partial\) operator by \textit{L. Hörmander} [Acta Math. 113, 89- 152 (1965; Zbl 0158.110); and An introduction to complex analysis in several variables, 2nd ed., North-Holland (1973; Zbl 0271.32001)] and proves fundamental theorems used in this paper. Two realizations of hyperfunctions (or generalized functions) can be found in several works as \textit{P. Schapira} [Théorie des hyperfonctions, Springer (1970; Zbl 0192.473)], \textit{H. Komatsu} [`An introduction to the theory of hyperfunctions', Iwanami (1978) (in Japanese)] and others. But self- awakening description of the equivalence and mutual independence of two realizations is first appeared in this paper. He defines the Fourier transformation of \(H\)-valued Fourier hyperfunctions and shows that the space of \(H\)-valued Fourier hyperfunctions on the entire space is stable under the Fourier transformation. He also proves the Paley-Wiener theorem for \(H\)-valued Fourier hyperfunctions. Thereby we have obtained the completed new version of the theory of (vector valued) Fourier hyperfunctions.