Microlocal analysis of double fibration transforms with conjugate points

TL;DR

This paper analyzes the structure of normal operators in double fibration transforms with conjugate points, revealing their decomposition into elliptic pseudodifferential and Fourier integral operators under certain conditions.

Contribution

It extends previous work by providing a microlocal analysis of these transforms, especially regarding the impact of conjugate points on the operator structure.

Findings

Normal operators split into elliptic pseudodifferential and Fourier integral parts.

Results depend on the distribution and degree of conjugate points.

Builds on and generalizes earlier studies of geodesic X-ray transforms.

Abstract

We study the structure of normal operators of double fibration transforms with conjugate points. Examples of double fibration transforms include Radon transforms, $d$-plane transforms on the Euclidean space, geodesic X-ray transforms, light-ray transforms, and ray transforms defined by null bicharacteristics associated with real principal type operators. We show that, under certain stable conditions on the distribution of conjugate points, the normal operator splits into an elliptic pseudodifferential operator and several Fourier integral operators, depending on the degree of the conjugate points. These problems were first studied for geodesic X-ray transforms by Stefanov and Uhlmann (Analysis \& PDE, {\bf 5} (2012), pp.219--260). After that Holman and Uhlmann (Journal of Differential Geometry, {\bf 108} (2018), pp.459--494) proved refined results according to the degree of regular conjugate points.