A quasianalytic singular spectrum with respect to the Denjoy-Carleman class

Abstract

Abstract Making use of the FBI (Fourier-Bros-Iagolnitzer) transforms we simplify the quasianalytic singular spectrum for the Fourier hyperfunctions, which was defined for distributions by Hörmander as follows; for any Fourier hyperfunction u , ( x 0 , ξ 0 ) does not belong to the quasianalytic singular spectrum W F M (u) if and only if there exist positive constants C, γ and N , and a neighborhood of x 0 and a conic neighborhood Г of ξ 0 such that for all x ∈ U , |ξ| ∈ Γ and |ξ| ≥ N , where M(t) is the associated function of the defining sequence M p . This result simplifies Hörmander’s definition and unify the singular spectra for the C ∞ class, the analytic class and the Denjoy-Carleman class, both quasianalytic and nonquasianalytic.