On Maximal Functions Associated to Families of Curves in the Plane

TL;DR

This paper develops a unified framework to analyze the $L^p$ bounds of maximal functions related to families of curves in the plane, including Kakeya and circular maximal functions, with applications to geometric measure theory.

Contribution

It introduces a new approach that unifies the study of various planar maximal functions and establishes sharp $L^p$ bounds using incidence geometry techniques.

Findings

Established sharp $L^p$ bounds for maximal functions of curves

Unified framework for different families of planar maximal functions

Applications to Furstenberg sets and projection problems

Abstract

We consider the $L^p$ mapping properties of maximal averages associated to families of curves, and thickened curves, in the plane. These include the (planar) Kakeya maximal function, the circular maximal functions of Wolff and Bourgain, and their multi-parameter analogues. We propose a framework that allows for a unified study of such maximal functions, and prove sharp $L^p\to L^p$ operator bounds in this setting. A key ingredient is an estimate from discretized incidence geometry that controls the number of higher order approximate tangencies spanned by a collection of plane curves. We discuss applications to the F\"assler-Orponen restricted projection problem, and the dimension of Furstenberg-type sets associated to families of curves.