Improved $L^p$ bounds for the helical maximal function in dimensions $n\ge 5$

Let $n\ge 5$. We prove that the helical maximal operator is bounded on $L^p({\mathbb{R}}^n)$ for some $p<2n-4$. This improves the previously known range $p>2n-4$ obtained by Gan-Maldague-Oh. As observed by Beltran-Hickman, the exponent $2n-4$ is the natural limit of the standard local smoothing method.