A natural 3-dimensional analogue of Bourgainâ€™s circular maximal function theorem
in the plane is the study of the sharp $L^p$ bounds in $\mathbb{R}^3$ for the
maximal function associated with averages over dilates of the helix (or, more
generally, of any curve with non-vanishing curvature and torsion). In this talk,
we present a sharp result, which establishes that $L^p$ bounds hold if and only
if $p>3$. This is joint work with Shaoming Guo, Jonathan Hickman and Andreas Seeger.

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Last updated: 17 May 2024