\(L^2\) Estimates for a Nikodym Maximal Function Associated to Space Curves

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