On the Wolff circular maximal function

Abstract

We prove sharp $L^{3}$ bounds for a variable coefficient generalization of the Wolff circular maximal function $M^{\delta}f$. For each fixed radius $r$, $M^{\delta}f(r)$ is the maximal average of $f$ over the $\delta$-neighborhood of a circle of radius $r$ and arbitrary center. In this paper, we consider maximal averages over families of curves satisfying the cinematic curvature condition, which was first introduced by Sogge to generalize the Bourgain circular maximal function. Our proof manages to avoid a key technical lemma in Wolff’s original argument, and thus our arguments also yield a shorter proof of the boundedness of the (conventional) Wolff circular maximal function. At the heart of the proof is an induction argument that employs an efficient partitioning of $\mathbb{R}^{3}$ into cells using the discrete polynomial ham sandwich theorem.