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On Maximal Functions Associated to Families of Curves in the Plane

Joshua Zahl

TL;DR

This work develops a unified L^p theory for maximal averages over families of plane curves by introducing an s-parameter framework tied to m-dimensional cinematic curve families. The core technical advance is a Kakeya-type incidence bound for higher-order tangencies, expressed via tangency rectangles and their prisms, established through polynomial partitioning and real-algebraic/ODE methods. This leads to sharp L^p bounds for maximal averages (Kakeya-type and Bourgain-type settings) and, through local smoothing frameworks, to translation-invariant maximal bounds and restricted-projection results. The approach yields geometric-measure-theoretic consequences, including dimension bounds for Furstenberg-type curve sets and generalized Furstenberg sets, thereby linking incidence geometry, harmonic analysis, and geometric measure theory in a cohesive framework.

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 Fässler-Orponen restricted projection problem, and the dimension of Furstenberg-type sets associated to families of curves.

On Maximal Functions Associated to Families of Curves in the Plane

TL;DR

This work develops a unified L^p theory for maximal averages over families of plane curves by introducing an s-parameter framework tied to m-dimensional cinematic curve families. The core technical advance is a Kakeya-type incidence bound for higher-order tangencies, expressed via tangency rectangles and their prisms, established through polynomial partitioning and real-algebraic/ODE methods. This leads to sharp L^p bounds for maximal averages (Kakeya-type and Bourgain-type settings) and, through local smoothing frameworks, to translation-invariant maximal bounds and restricted-projection results. The approach yields geometric-measure-theoretic consequences, including dimension bounds for Furstenberg-type curve sets and generalized Furstenberg sets, thereby linking incidence geometry, harmonic analysis, and geometric measure theory in a cohesive framework.

Abstract

We consider the 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 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 Fässler-Orponen restricted projection problem, and the dimension of Furstenberg-type sets associated to families of curves.
Paper Structure (32 sections, 39 theorems, 280 equations)

This paper contains 32 sections, 39 theorems, 280 equations.

Table of Contents

  1. Introduction
  2. The Setup
  3. Kakeya-type maximal functions
  4. Bourgain-type maximal functions
  5. An $L^p$ estimate for collections of plane curves
  6. Applications to geometric measure theory
  7. Curve tangencies, and tangency rectangles
  8. Main ideas, and a sketch of the proof
  9. Paper organization
  10. Thanks
  11. Notation
  12. Curves and tangency rectangles
  13. Initial reductions
  14. Tangency Rectangles and Tangency prisms
  15. Tangency and rescaling
  16. ...and 17 more sections

Key Result

Theorem 1.5

Let $m>s\geq 1$ be integers, and let $M_\delta$ be an $s$-parameter maximal function associated to an $m$-dimensional family of cinematic curves. Let $\varepsilon>0$. Then for all $\delta>0$ sufficiently small, we have

Theorems & Definitions (91)

  • Definition 1.1
  • Example 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5
  • Definition 1.6
  • Definition 1.7
  • Theorem 1.8
  • Definition 1.9
  • Example 1.10
  • ...and 81 more