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Unique continuation for the momentum ray transform

Joonas Ilmavirta, Pu-Zhao Kow, Suman Kumar Sahoo

Abstract

The present article focuses on a unique continuation result for certain weighted ray transforms, utilizing the unique continuation property (UCP) of the fractional Laplace operator. Specifically, we demonstrate a conservative property for momentum ray transforms acting on tensors, as well as the antilocality property for both weighted ray and cone transforms acting on functions.

Unique continuation for the momentum ray transform

Abstract

The present article focuses on a unique continuation result for certain weighted ray transforms, utilizing the unique continuation property (UCP) of the fractional Laplace operator. Specifically, we demonstrate a conservative property for momentum ray transforms acting on tensors, as well as the antilocality property for both weighted ray and cone transforms acting on functions.
Paper Structure (16 sections, 20 theorems, 126 equations)

This paper contains 16 sections, 20 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Momentum ray transform on Schwartz space
  4. Momentum ray transform on compactly supported tensor field distributions
  5. Normal operator of momentum ray transforms
  6. Some differential operators
  7. Generalized Saint-Venant operator
  8. A generalization of the curl-curl identity
  9. A solenoidal decomposition theorem
  10. Main results
  11. Unique continuation property for momentum ray transform on tensors
  12. Measurable unique continuation property of momentum ray transform
  13. Generalization of momentum ray transform: Fractional momentum ray transform
  14. Antilocality property of the weighted ray transform
  15. Antilocality property of the cone transform
  16. ...and 1 more sections

Key Result

Theorem I

Let $m \in \mathbb{N}$ and an integer $0 \le k \le m$. Suppose that and assume that there exists an nonempty open set $U$ in $\mathbb{R}^{n}$ such that If there exists $x_{0} \in U$ such that for each $0 \leq p \leq k$ the following assumption holds: then $R^{k}f \equiv 0$ in $\mathbb{R}^{n}$. If we additionally assuming that $f$ has compact support, then $f$ is a generalized potential field, t

Theorems & Definitions (48)

  • Theorem I
  • Theorem II
  • Theorem III
  • Definition 2.1
  • Remark 2.2: An equivalent relation
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Remark 2.5: An equivalent definition
  • Remark 2.6
  • ...and 38 more