Sharp Sobolev regularity of restricted X-ray transforms

TL;DR

This paper establishes sharp Sobolev regularity estimates for restricted X-ray transforms generated by nondegenerate curves in higher dimensions, extending previous results from three dimensions to all dimensions greater than or equal to three.

Contribution

The authors extend sharp $L^p$-Sobolev regularity estimates for restricted X-ray transforms from three dimensions to all higher dimensions using an inductive strategy.

Findings

Established sharp $L^p$-Sobolev regularity estimates in $ ext{R}^{d+1}$ for $d geq 3$

Extended previous 3D results to higher dimensions

Utilized inductive approach for the proof

Abstract

We study $L^p$-Sobolev regularity estimate for the restricted X-ray transforms generated by nondegenerate curves. Making use of the inductive strategy in the recent work by the authors, we establish the sharp $L^p$-regularity estimates for the restricted X-ray transforms in $\mathbb R^{d+1}$, $d\ge 3$. This extends the result due to Pramanik and Seeger in $\mathbb R^3$ to every dimension.