On localization properties of Fourier transforms of hyperfunctions

In [A.G. Smirnov, Fourier transformation of Satoʹs hyperfunctions, Adv. Math. 196 (2005) 310–345] the author introduced a new generalized function space U ( R k ) which can be naturally interpreted as the Fourier transform of the space of Satoʹs hyperfunctions on R k . It was shown that all Gelfand–Shilov spaces S α ′ 0 ( R k ) ( α > 1 ) of analytic functionals are canonically embedded in U ( R k ) . While the usual definition of support of a generalized function is inapplicable to elements of S α ′ 0 ( R k ) and U ( R k ) , their localization properties can 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 α ′ 0 ( R k ) and U ( R k ) is studied. It is proved that an analytic functional u ∈ S α ′ 0 ( R k ) is carried by a cone K ⊂ R k if and only if its canonical image in U ( R k ) is carried by K.