On localization properties of Fourier transforms of hyperfunctions

Abstract

In [Adv. Math. 196 (2005) 310-345] the author introduced a new generalized function space $\mathcal{U}(R^k)$ which can be naturally interpreted as the Fourier transform of the space of Sato's hyperfunctions on $R^k$. It was shown that all Gelfand--Shilov spaces $S_{\alpha }^{\prime 0}(R^k)$ ($\alpha >1$) of analytic functionals are canonically embedded in $\mathcal{U}(R^k)$. While the usual definition of support of a generalized function is inapplicable to elements of $S_{\alpha }^{\prime 0}(R^k)$ and $\mathcal{U}(R^k)$, their localization properties can be consistently described using the concept of {\it 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 $S_{\alpha }^{\prime 0}(R^k)$ and $\mathcal{U}(R^k)$ is studied. It is proved that an analytic functional $u\in S_{\alpha }^{\prime 0}(R^k)$ is carried by a cone $K\subset R^k$ if and only if its canonical image in $\mathcal{U}(R^k)$ is carried by $K$.

21 pages, final version, accepted for publication in J. Math. Anal. Appl