$L^2$ estimates for a Nikodym maximal function associated to space curves

We consider the $L^p\to L^p$ boundedness of a Nikodym maximal function associated to a one-parameter family of tubes in ${\mathbb{R}}^{d+1}$ whose directions are determined by a non-degenerate curve $\gamma$ in ${\mathbb{R}}^d$. These operators arise in the analysis of maximal averages over space curves. The main theorem generalises the known results for $d=2$ and $d=3$ to general dimensions. The key ingredient is an induction scheme motivated by recent work of Ko–Lee–Oh.