Sharp Sobolev regularity of restricted X-ray transforms

Abstract

We study $L^p$ -Sobolev regularity estimates 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 $\mathbb {R}^3$ .