Sharp helical maximal estimates in all dimensions

Abstract

We establish sharp $L^p$ maximal estimates for averages over smooth nondegenerate curves in $\mathbb R^n$ for $p>n$. More precisely, we show that for any nondegenerate curve $γ$ in $\mathbb R^n$ and the associated maximal operator \begin{equation} M f(x):=\sup_{t>0} \Big|\int_{-1}^1f(x-tγ(s))χ(s)\,\mathrm{d}s\Big|, \end{equation} the estimate \begin{equation} \| Mf\|_{L^p(\mathbb R^n)}\lesssim_{γ,p,n} \|f\|_{L^p(\mathbb R^n)} \end{equation} holds for all $p>n$. A new ingredient is a weighted Kakeya-type estimate.