A note on restricted X-ray transforms

TL;DR

This paper applies Christ's techniques to establish endpoint $L^p-L^q$ bounds and Lorentz space estimates for the X-ray transform linked to a specific polynomial curve in higher dimensions.

Contribution

It introduces a novel application of Christ's methods to derive endpoint bounds for a class of X-ray transforms associated with polynomial curves.

Findings

Established endpoint $L^p-L^q$ bounds for the X-ray transform

Derived almost-sharp Lorentz space estimates

Extended techniques to polynomial curve-generated line complexes

Abstract

We show how the techniques introduces by Christ can be employed to derive endpoint $L^p-L^q$ bounds for the X-ray transform associated to the line complex generated by the curve $t\to(t,t^2,...,t^{d-1}).$ Almost-sharp Lorentz space estimates are produced as well.