Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Ray transform on Sobolev spaces of symmetric tensor fields, II: Range characterization

Venky P. Krishnan, Vladimir A. Sharafutdinov

Abstract

The ray transform $I$ integrates symmetric $m$-tensor field in $\mathbb{R}^n$ over lines. This transform in Sobolev spaces was studied in our earlier work where higher order Reshetnyak formulas (isometry relations) were established. The main focus of the current work is the range characterization. In dimensions $n\geq 3$, the range characterization of the ray transform in Schwartz spaces is well-known; the main ingredient of the characterization is a system of linear differential equations of order $2(m+1)$ which are called John equations. Using the higher order Reshetnyak formulas, the range of the ray transform on Sobolev spaces is characterized in dimensions $n\geq 3$ in this paper.

Ray transform on Sobolev spaces of symmetric tensor fields, II: Range characterization

Abstract

The ray transform integrates symmetric -tensor field in over lines. This transform in Sobolev spaces was studied in our earlier work where higher order Reshetnyak formulas (isometry relations) were established. The main focus of the current work is the range characterization. In dimensions , the range characterization of the ray transform in Schwartz spaces is well-known; the main ingredient of the characterization is a system of linear differential equations of order which are called John equations. Using the higher order Reshetnyak formulas, the range of the ray transform on Sobolev spaces is characterized in dimensions in this paper.
Paper Structure (9 sections, 10 theorems, 129 equations)

This paper contains 9 sections, 10 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and statement of the main result
  3. Ray transform of symmetric tensor fields
  4. Fourier slice theorem
  5. Range characterization of the ray transform on the Schwartz space
  6. The vector fields $X_i$ and $\Xi_i$ and intrinsic John operators
  7. The spaces $H^{(r,s)}_t(T\mathbb{S}^{n-1})$
  8. Range characterization for ray transform of Schwartz class tensor fields using intrinsic John operators
  9. Proof of Lemma \ref{['L3.3']}

Key Result

Theorem 2.1

A function $\varphi\in{\mathcal{S}}(T{\mathbb S}^{n-1})\ (n\geq3)$ belongs to the range of the operator 1.1 if and only if the following two conditions hold:

Theorems & Definitions (19)

  • Theorem 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Theorem 2.4: Main theorem
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Corollary 3.3
  • ...and 9 more