On traces of Fourier integral operators localized at a finite set of points

TL;DR

This paper investigates the behavior of Fourier integral operators when restricted to a submanifold, especially focusing on cases where the resulting trace operator is localized at finitely many points, revealing specific structural properties.

Contribution

It characterizes the traces of Fourier integral operators localized at finite points, providing new insights into their structure and behavior in the context of embedded manifolds.

Findings

Trace operators can be expressed as Fourier--Mellin operators under certain conditions.

Localization at finite points leads to specific structural forms of the trace.

The results connect Fourier integral operators with localized Fourier--Mellin operators.

Abstract

Given a smooth embedding of manifolds $i: X \hookrightarrow M$ and a Fourier integral operator $\Phi$ acting on $M$, obtained by quantization of a canonical transformation, consider its trace $i^!(\Phi)$ on $X$ (in the sense of relative theory). We discuss the situation when $i^!(\Phi)$ has the form of a Fourier--Mellin operator and, in particular, is localized at a finite set of points.