Sharp $L^p$ range for the helical maximal operator

Determine whether the helical maximal operator associated with a smooth nondegenerate curve in $mathbb{R}^n$ is bounded on $L^p(mathbb{R}^n)$ for every $p>n$, thereby establishing whether $p>n$ is the sharp range of exponents.

Background

The paper studies the helical maximal operator $\mathscr Mf(x)=\sup_{t>0}|A_tf(x)|$, where $A_t$ averages a function over dilates of a smooth nondegenerate curve in $\mathbb{R}^n$. The main theorem proves boundedness for $p>2n-4-\epsilon_0$ when $n\geq5$, improving the previously known range but not reaching the conjectured optimal threshold.

A standard example gives the necessary condition $p>n$. The authors explicitly state that this necessary condition is conjectured to be sufficient, leaving the sharp maximal estimate unresolved, particularly in dimensions $n\geq5$.