On traces of Fourier integral operators on submanifolds

TL;DR

This paper investigates the behavior of Fourier integral operators when restricted to submanifolds, establishing conditions for their traces to remain Fourier integral operators and computing their amplitudes.

Contribution

It provides new criteria for the trace of Fourier integral operators on submanifolds to also be Fourier integral operators and explicitly calculates their amplitudes.

Findings

Conditions under which the trace remains a Fourier integral operator

Explicit formulas for the amplitude of the trace operator

Extension of Fourier integral operator theory to submanifold traces

Abstract

Given a smooth embedding $i\colon X \hookrightarrow M$ of manifolds and a Fourier integral operator $\Phi = \Phi(\Lambda)$ on $M$ associated with a Lagrangian submanifold $\Lambda \subset T^*(X\times X) \setminus \{0\}$, we consider its trace $i^!(\Lambda)$ on the submanifold $X$, i.e. the composition $i^* \Phi i_*$, where $i^*$ and $i_*$ are the boundary and coboundary operators, respectively. We establish the conditions under which the trace $i^!(\Phi)$ is also a Fourier integral operator and calculate its amplitude in canonical local coordinates.