Compactness of a restricted X-ray transform

TL;DR

This paper proves the existence and precompactness of extremizers for a restricted X-ray transform along the moment curve, using advanced Lorentz space methods in a mixed norm setting.

Contribution

It introduces a novel approach extending Christ’s Lorentz space method to mixed norm Lebesgue spaces for analyzing the restricted X-ray transform.

Findings

Existence of extremizers for the restricted X-ray transform.

Precompactness of extremizing sequences modulo symmetry.

Extension of Lorentz space techniques to mixed norm spaces.

Abstract

We show that the X-ray transform with directions restricted along the moment curve possesses extremizers and that $L^p$-normalized extremizing sequences are precompact modulo symmetry. Our approach advances the Lorentz space method of Christ to a mixed norm Lebesgue space setting.