Microlocal analysis of the light ray transform on globally hyperbolic Lorentzian manifolds

TL;DR

This paper analyzes the light ray transform on globally hyperbolic Lorentzian manifolds, revealing the structure of its normal operator and providing Sobolev estimates to understand light-like singularities.

Contribution

It characterizes the Schwartz kernel of the normal operator as a paired Lagrangian distribution with non-vanishing symbols and clarifies the determination of light-like singularities.

Findings

Normal operator's Schwartz kernel is a paired Lagrangian distribution.

Established Sobolev estimates for the light ray transform.

Clarified how light-like singularities are determined by the normal operator.

Abstract

For the light ray transform on globally hyperbolic Lorentzian manifolds of dimension $n+1 \geq 3$ acting on compactly supported distributions, we show that the Schwartz kernel of the normal operator is a paired Lagrangian distribution with non-vanishing principal symbols on each Lagrangians. We obtain Sobolev estimates for the light ray transform, and clarify the determination of light-like singularities using the normal operator.