Wigner Analysis of Fourier Integral Operators with symbols in the Shubin classes

2024/02/05 by Elena Cordero, Gianluca Giacchi, Cordero, Elena +5 · 2 citations
Mathematics · #35S05 #35S30 #42C15 #47G30 #Algebraic and Geometric Analysis #Differential Equations and Boundary Problems #FOS: Mathematics #Functional Analysis (math.FA) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2402.02809

openalex publication_date 2024/02/05 · openalex created_date 2024/02/08 · openalex updated_date 2026/07/28

Abstract

We study the decay properties of Wigner kernels for Fourier integral operators of types I and II. The symbol spaces that allow a nice decay of these kernels are the Shubin classes Γ^m(\mathbbR^2d), with negative order m. The phases considered are the so-called tame ones, which appear in the Schrödinger propagators. The related canonical transformations are allowed to be nonlinear. It is the nonlinearity of these transformations that are the main obstacles for nice kernel localizations when symbols are taken in the Hörmander's class S⁰_0,0(\mathbbR^2d). Here we prove that Shubin classes overcome this problem and allow a nice kernel localization, which improves with the decreasing of the order m.

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• Resolvent trace asymptotics for operators in the Shubin class
• A Characterization of Metaplectic Time-Frequency Representations

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