On the Wolff circular maximal function

TL;DR

This paper establishes sharp $L^3$ bounds for a generalized Wolff circular maximal function involving variable curves satisfying the cinematic curvature condition, using an innovative polynomial partitioning approach.

Contribution

It provides a new proof avoiding a key lemma in Wolff's original argument and extends the bounds to variable coefficient settings with cinematic curvature.

Findings

Proved sharp $L^3$ bounds for the generalized maximal function.

Developed a shorter proof for the classical Wolff circular maximal function.

Introduced an efficient polynomial partitioning method in the proof.

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 proof also yields 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.