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Microlocal analysis of double fibration transforms with conjugate points

Hiroyuki Chihara

TL;DR

This paper extends microlocal analysis of double fibration transforms to accommodate conjugate points. It shows that when no conjugate triplets occur, the normal operator $\mathcal{R}^*\mathcal{R}$ is an elliptic pseudodifferential operator, enabling inversion up to smoothing. When regular conjugate points are present, $\mathcal{R}^*\mathcal{R}$ decomposes into an elliptic PDO part plus a finite family of Fourier integral operators whose orders depend on the degree of conjugacy, with canonical relations determined by the conjugate geometry. The results generalize the Holman--Uhlmann framework to the broader setting of double fibrations and provide a structured decomposition that informs stability and reconstruction in inverse problems from integral geometry.

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.

Microlocal analysis of double fibration transforms with conjugate points

TL;DR

This paper extends microlocal analysis of double fibration transforms to accommodate conjugate points. It shows that when no conjugate triplets occur, the normal operator is an elliptic pseudodifferential operator, enabling inversion up to smoothing. When regular conjugate points are present, decomposes into an elliptic PDO part plus a finite family of Fourier integral operators whose orders depend on the degree of conjugacy, with canonical relations determined by the conjugate geometry. The results generalize the Holman--Uhlmann framework to the broader setting of double fibrations and provide a structured decomposition that informs stability and reconstruction in inverse problems from integral geometry.

Abstract

We study the structure of normal operators of double fibration transforms with conjugate points. Examples of double fibration transforms include Radon transforms, -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.

Paper Structure

This paper contains 4 sections, 13 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Normal operators without conjugate points
  4. Normal operators with conjugate points

Key Result

Theorem 1

For $f(x)=\mathcal{O}(\langle{x}\rangle^{-d-\varepsilon})$ on $\mathbb{R}^n$ for some $\varepsilon>0$, we have where $x=(x_1,\dotsc,x_n)\in\mathbb{R}^n$, $\langle{x}\rangle:=\sqrt{1+x_1^2+\dotsb+x_n^2}$, $-\Delta_{\mathbb{R}^n}:=-\partial_{x_1}^2-\dotsb-\partial_{x_n}^2$.

Theorems & Definitions (20)

  • Theorem 1: Helgason
  • Theorem 2: HolmanUhlmann
  • Theorem 3: MazzucchelliSaloTzou and Hoermander4
  • Lemma 4: MazzucchelliSaloTzou
  • Lemma 4: MazzucchelliSaloTzou
  • Lemma 5
  • proof
  • Definition 6
  • Lemma 7: MazzucchelliSaloTzou
  • Lemma 8
  • ...and 10 more