Lagrangian distributions and Fourier integral operators with quadratic phase functions and Shubin amplitudes

TL;DR

This paper investigates Fourier integral operators with quadratic phase functions and Shubin amplitudes, providing a factorization approach and a new class of Lagrangian distributions related to twisted graph Lagrangians.

Contribution

It introduces a factorization of these operators into Weyl pseudodifferential and metaplectic operators, and defines a new class of Lagrangian tempered distributions.

Findings

Operators can be factorized into Weyl and metaplectic components.

Characterization of Schwartz kernels via phase space estimates.

Kernels are equivalent to Lagrangian distributions with twisted graph Lagrangians.

Abstract

We study Fourier integral operators with Shubin amplitudes and quadratic phase functions associated to twisted graph Lagrangians with respect to symplectic matrices. We factorize such an operator as the composition of a Weyl pseudodifferential operator and a metaplectic operator and derive a characterization of its Schwartz kernel in terms of phase space estimates. Extending the conormal distributions in the Shubin calculus, we define an adapted notion of Lagrangian tempered distribution. We show that the kernels of Fourier integral operators are identical to Lagrangian distributions with respect to twisted graph Lagrangians.