Momentum Ray Transforms, II: Range Characterization In the Schwartz space

TL;DR

This paper characterizes the range of momentum ray transforms of tensor fields in the Schwartz space, generalizing classical equations and conditions for dimensions three and two respectively.

Contribution

It provides a comprehensive range characterization for the momentum ray transform on Schwartz tensor fields, extending classical results to higher orders and dimensions.

Findings

Range characterized by differential equations in n≥3 dimensions.

Range characterized by integral conditions in 2 dimensions.

Generalization of John and Gelfand--Helgason--Ludwig conditions.

Abstract

The momentum ray transform $I^k$ integrates a rank $m$ symmetric tensor field $f$ over lines of ${\R}^n$ with the weight $t^k$: $ (I^k\!f)(x,\xi)=\int_{-\infty}^\infty t^k\l f(x+t\xi),\xi^m\r\,dt. $ We give the range characterization for the operator $f\mapsto(I^0\!f,I^1\!f,\dots, I^m\!f)$ on the Schwartz space of rank $m$ smooth fast decaying tensor fields. In dimensions $n\ge3$, the range is characterized by certain differential equations of order $2(m+1)$ which generalize the classical John equations. In the two-dimensional case, the range is characterized by certain integral conditions which generalize the classical Gelfand -- Helgason -- Ludwig conditions.