Continuity of Gevrey–Hörmander pseudo-differential operators on modulation spaces

Abstract

Let s ≥ 1, ω, ω 0 ∈ P 0 E,s , a ∈ Γ (ω0) s , and let B be a suitable invariant quasi-Banach function space, Then we prove that the pseudo-differential operator Op(a) is continuous from M (ω 0 ω, B) to M (ω, B).(0.1) However, by replacing the classical function and distribution spaces with suitable Gelfand-Shilov or Gevrey spaces and their spaces of ultradistributions, the problem (0.1) become well-posed. (See [3, 25].) An other classical example concerns the heat problemwhere Ω is a cuboid. It is well-posed when moving forward in time (t > 0), but not well-posed when moving backwards in time (t < 0)2010 Mathematics Subject Classification. 35S05, 47B37, 47G30, 42B35.

Anisotropic Gevrey-Hörmander Pseudo-Differential Operators on Modulation Spaces

“…Similar investigations were performed in [43] in the case s = σ (i. e. the isotropic case). Therefore, the results in the current paper are more general in the sense of the anisotropicity of the considered symbol classes.…”

“…Lemma 2.4 follows by similar arguments as in [43]. In order to be self contained we give a different proof.…”

“…We notice that M(ω, B) is independent of the choice of φ in Definition 1.8 cf. [43]. Furthermore, M(ω, B) is a Banach space in view of [28].…”

“…Therefore, the results in the current paper are more general in the sense of the anisotropicity of the considered symbol classes. Moreover, we use different techniques compared to [43].…”

“…The smoothness of H is a consequence of the uniqueness of the adjoint (cf. Remark 2.2) and[43, Lemma 2.7].To show that (2.7) holds, letΦ 0 (x, ξ, z, ζ) = Φ(x, ξ, z, ζ)φ 2 (z),where Φ defined as in(2.4), and let Ψ = F 3 Φ 0 , where F 3 Φ 0 is the partial Fourier transform of Φ 0 (x, ξ, z, ζ) with respect to the z variable. Then it follows from the assumptions and (2.3) ′ that…”