On localization properties of Fourier transforms of hyperfunctions

In [Adv. Math. 196 (2005) 310–345] the author introduced a new generalized function space U(Rk) which can 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′0 α (R k) (α > 1) of analytic functionals are canonically embedded in U(Rk). While the usual definition of support of a generalized function is inapplicable to elements of S′0 α (R k) and U(Rk), their localization properties can be consistently described using the concept of carrier cone introduced by Soloviev [Lett. Math. Phys. 33 (1995) 49–59; Comm. Math. Phys. 184 (1997) 579–596]. In this paper, the relation between carrier cones of elements of S′0 α (R k) and U(Rk) is studied. It is proved that an analytic functional u ∈ S′0 α (R k) is carried by a cone K ⊂ Rk if and only if its canonical image in U(Rk) is carried by K.