Theory of $H$-valued Fourier hyperfunctions

Fourier transformation of Sato’s hyperfunctions

A new generalized function space in which all Gelfand-Shilov classes S ′0 α (α > 1) of analytic functionals are embedded is introduced. This space of ultrafunctionals does not possess a natural nontrivial topology and cannot be obtained via duality from any test function space. A canonical isomorphism between the spaces of hyperfunctions and ultrafunctionals on R is constructed that extends the...

Structural Theorems for Families of Fourier Hyperfunctions

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 inappl...

Euclidean Reconstruction in Quantum Field Theory: between Tempered Distributions and Fourier Hyperfunctions

In this short note on my talk I want to point out the mathematical difficulties that arise in the study of the relation of Wightman and Euclidean quantum field theory, i.e., the relation between the hierarchies of Wightman and Schwinger functions. The two extreme cases where the reconstructed Wightman functions are either tempered distributions — the wellknown Osterwalder-Schrader reconstructio...

Right inverses for partial differential operators on Fourier hyperfunctions

We characterize the partial differential operators P (D) admitting a continuous linear right inverse in the space of Fourier hyperfunctions by means of a dual (Ω)-type estimate valid for the bounded holomorphic functions on the characteristic variety VP near R . The estimate can be transferred to plurisubharmonic functions and is equivalent to a uniform (local) Phragmén–Lindelöf-type condition.