A Characterization for Fourier Hyperfunctions

The space of test functions for Fourier hyperfunctions is characterized by two conditions sup| 0. Combining this result and the new characterization of Schwartz space in [1] we can easily compare two important spaces J and 3 which are both invariant under Fourier transformations. § 0. Introduction The purpose of this paper is to give new characterization of the space £F of test functions for the Fourier hyperfunctions. In [6], K. W. Kim, S. Y. Chung and D. Kim introduce the real version of the space EF of test functions for the Fourier hyperfunctions as follows, *- 0\. They also show the equivalence of the above definition and Sato-Kawai's original definition in complex form. Also, in [1] J. Chung, S. Y. Chung and D. Kim give new characterization of the Schwartz space S, i.e., show that for O) |exp&|%| 0. Observing the above growth conditions we can easily see that the space £F which is invariant under the Fourier transformation is much smaller than Communicated by T. Kawai, April 5, 1993. Revised May 20, 1993. 1991 Mathematics Subject Classification: 46F12, 46F15. * Department of Mathematics, Kunsan National University, Kunsan 573-360, Korea. ** Department of Mathematics, Duksung Women's University, Seoul 132-714, Korea. Department of Mathematics, Seoul National University, Seoul 151-742, Korea. Partially supported by the Ministry of Education and GARC. 204 JAEYOUNG CHUNG, SOON-YEONG CHUNG AND DOHAN KIM Schwartz space S. Since an element in the strong dual £F' of the space 3 is called a Fourier hyperfunction, the space 9"' of Fourier hyperfunctions which is also invariant under the Fourier transformation is much bigger than the space S of tempered distributions. Section 1 is devoted to providing the necessary definitions and preliminaries. We prove the main theorem in Section 2. § 1. Preliminaries We use the multi-index notations; for x = (xl} , xn\ f = (&, , fn)e/2 n and a multi-index a=(a1} az, , aJejV?, 8 =d^ dfi, \a =«H ----\-an with dj=d/dxjf and NQ the set of non-negative integers. For f^L(R] the Fourier transform / is the bounded continuous function in R defined by (1.1) Definition 1.1. We denote by (;c)i

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