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

Elena Cordero, Gianluca Giacchi, Luigi Rodino, Mario Valenzano

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 $Γ^m(\mathbb{R^{2d}})$, 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 $S^{0}_{0,0}(\mathbb{R^{2d}})$. Here we prove that Shubin classes overcome this problem and allow a nice kernel localization, which improves with the decreasing of the order $m$.

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

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 , with negative order . 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 . Here we prove that Shubin classes overcome this problem and allow a nice kernel localization, which improves with the decreasing of the order .
Paper Structure (9 sections, 14 theorems, 112 equations)

This paper contains 9 sections, 14 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The symplectic group $Sp(d,\mathbb{R})$, metaplectic operators and Wigner distributions
  4. Shubin and Hörmander classes shubin, helffer84, Elena-book
  5. Tame phase functions and related canonical transformations
  6. Properties of the Wigner Kernel
  7. Properties of FIO($\chi, N$)
  8. FIOs of type I
  9. FIOs of Type II

Key Result

Theorem 1.3

(i) Boundedness. $T\in FIO(\chi, N)$ is bounded on $L^2(\mathbb{R}^d)$. (ii) Algebra Property. If $T_i\in FIO(\chi_i,N)$, $i=1,2$, then $T_1 T_2\in FIO(\chi_1\chi_2,N)$. (iii) If $T\in FIO(\chi, N)$ then its adjoint $T^{*}$ is in $FIO(\chi^{-1}, N)$.

Theorems & Definitions (30)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3: Properties of the class FIO($\chi$, $N$)
  • Example 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5
  • Theorem 3.1
  • proof
  • ...and 20 more