Propagation of singularities around a Lagrangian submanifold of radial points

TL;DR

This paper investigates the propagation of singularities in solutions to pseudodifferential equations near a Lagrangian submanifold where the Hamiltonian vector field is radial, providing localized regularity results that extend previous global analyses.

Contribution

It introduces a microlocal approach to analyze singularity propagation at a specific point on a radial Lagrangian submanifold, refining earlier global results by Melrose and Vasy.

Findings

Achieves regularity at a point in the radial set under weaker assumptions.

Extends positive-commutator methods to localized settings.

Provides results applicable to scattering theory contexts.

Abstract

In this work we study the wavefront set of a solution u to Pu = f, where P is a pseudodifferential operator on a manifold with real-valued homogeneous principal symbol p, when the Hamilton vector field corresponding to p is radial on a Lagrangian submanifold contained in the characteristic set of P. The standard propagation of singularities theorem of Duistermaat-Hormander gives no information at the Lagrangian submanifold. By adapting the standard positive-commutator estimate proof of this theorem, we are able to conclude additional regularity at a point q in this radial set, assuming some regularity around this point. That is, the a priori assumption is either a weaker regularity assumption at q, or a regularity assumption near but not at q. Earlier results of Melrose and Vasy give a more global version of such analysis. Given some regularity assumptions around the Lagrangian submanifold, they obtain some regularity at the Lagrangian submanifold. This paper microlocalizes these results, assuming and concluding regularity only at a particular point of interest. We then proceed to prove an analogous result, useful in scattering theory, followed by analogous results in the context of Lagrangian regularity.