A note on restricted X-ray transforms

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 $t\to (t,t^2,\mathrm{.}\mathrm{.}\mathrm{.},t^{d-1})\mathrm{.}$ Almost-sharp Lorentz space estimates are produced as well.