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

Abstract

We consider the $L^p \rightarrow L^p$ 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.