On a Class of Fourier Integral Operators on Manifolds with Boundary

TL;DR

This paper develops a calculus for a class of Fourier integral operators on manifolds with boundary, focusing on symplectomorphisms that preserve the boundary, and establishes their continuity properties and local representations.

Contribution

It introduces a Boutet de Monvel type calculus for boundary-preserving Fourier integral operators and analyzes their local phase functions for simplified proofs.

Findings

Established continuity properties of the operators.

Provided local operator-valued symbol representations.

Analyzed properties of the local phase functions.

Abstract

We study a class of Fourier integral operators on compact manifolds with boundary, associated with a natural class of symplectomorphisms, namely, those which preserve the boundary. A calculus of Boutet de Monvel's type can be defined for such Fourier integral operators, and appropriate continuity properties established. One of the key features of this calculus is that the local representations of these operators are given by operator-valued symbols acting on Schwartz functions or temperate distributions. Here we focus on properties of the corresponding local phase functions, which allow to prove this result in a rather straightforward way.