Injectivity and range description of first $(k+1)$ integral moment transforms over $m$-tensor fields in $\mathbb{R}^n$

TL;DR

This paper introduces a new tensor field decomposition, establishes injectivity of certain integral transforms, and characterizes their range using John's equation, advancing the mathematical understanding of tensor tomography.

Contribution

It generalizes tensor decomposition and provides new injectivity and range results for integral moment transforms of tensor fields in Euclidean space.

Findings

New tensor decomposition generalizing solenoidal and potential parts

Injectivity of first $(k+1)$ integral moment transforms

Range characterization via John's equation

Abstract

In this work, we prove a new decomposition result for rank $m$ symmetric tensor fields which generalizes the well known solenoidal and potential decomposition of tensor fields. This decomposition is then used to describe the kernel and to prove an injectivity result for first $(k+1)$ integral moment transforms of symmetric $m$-tensor fields in $\mathbb{R}^n$. Additionally, we also present a range characterization for first $(k+1)$ integral moment transforms in terms of the John's equation.