On localization properties of Fourier transforms of hyperfunctions

In [A.G. Smirnov, Fourier transformation of Sato's hyperfunctions, Adv. Math. 196 (2005) 310-345] the author introduced a new generalized function space u(R(k)) which Call be naturally interpreted as the Fourier transform of the space of Sato's hyperfunctions on Rk. It was shown that all Gelfand-Shilov spaces S(alpha)'(0)(Rk) (alpha > 1) of analytic functionals are canonically embedded in U(R). While the Usual definition of support of a generalized function is inapplicable to elements of S(alpha)'(0)(Rk) and u(R(k)), their localization properties call be consistently described using the concept of carrier cone introduced by Soloviev [M.A. Soloviev, Towards a generalized distribution formalism for gauge quantum fields, Lett. Math. Phys. 33 (1995) 49-59; M.A. Soloviev. An extension of distribution theory and of the Paley-Wiener-Schwartz theorem related to quantum gauge theory, Comm. Math. Phys. 184 (1997) 579-596]. In this paper, the relation between carrier cones of elements of S(alpha)'(0)(R(k)) and u(R(k)) is Studied. It is proved that an analytic functional u is an element of u(R(k)) is carried by a cone K subset of R(k) if and only if its canonical image in u(R(k)) is carried by K. (c) 2008 Elsevier Inc. All rights reserved.