On the Wolff circular maximal function

Details On the Wolff circular maximal function On the Wolff circular maximal function

2011-10-25

arxiv.org/abs/1110.5595v1

Mathematics

Classical Analysis and ODEs

10 pages

Scientific paper

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.

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