A normal form around a Lagrangian submanifold of radial points

TL;DR

This paper develops microlocal normal forms for pseudodifferential operators with a Lagrangian submanifold of radial points, advancing understanding of their classical dynamics and setting the stage for further parametrix and Poisson operator constructions.

Contribution

It introduces a microlocal normal form for a class of pseudodifferential operators with radial points on a Lagrangian submanifold, addressing a key gap in the analysis.

Findings

Provides a microlocal normal form for operators with radial points

Answers questions about classical dynamics near radial points

Lays groundwork for constructing parametrices and Poisson operators

Abstract

In this work we produce microlocal normal forms for pseudodifferential operators which have a Lagrangian submanifold of radial points. This answers natural questions about such operators and their associated classical dynamics. In a sequel, we will give a microlocal parametrix construction, as well as a construction of a microlocal Poisson operator, for such pseudodifferential operators.