$$L^2$$ Estimates for a Nikodym Maximal Function Associated to Space Curves

Abstract

For \(p \in [2,\infty )\), 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.