  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.
