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Maximal directional operators along algebraic varieties

Francesco Di Plinio, Ioannis Parissis

Abstract

We establish the sharp growth order, up to epsilon losses, of the $L^2$-norm of the maximal directional averaging operator along a finite subset $V$ of a polynomial variety of arbitrary dimension $m$, in terms of cardinality. This is an extension of the works by Córdoba, for one-dimensional manifolds, Katz for the circle in two dimensions, and Demeter for the 2-sphere. For the case of directions on the two-dimensional sphere we improve by a factor of $\sqrt{\log N}$ on the best known bound, due to Demeter, and we obtain a sharp estimate for our model operator. Our results imply new $L^2$-estimates for Kakeya-type maximal functions with tubes pointing along polynomial directions. Our proof technique is novel and in particular incorporates an iterated scheme of polynomial partitioning on varieties adapted to directional operators, in the vein of Guth, Guth-Katz, and Zahl.

Maximal directional operators along algebraic varieties

Abstract

We establish the sharp growth order, up to epsilon losses, of the -norm of the maximal directional averaging operator along a finite subset of a polynomial variety of arbitrary dimension , in terms of cardinality. This is an extension of the works by Córdoba, for one-dimensional manifolds, Katz for the circle in two dimensions, and Demeter for the 2-sphere. For the case of directions on the two-dimensional sphere we improve by a factor of on the best known bound, due to Demeter, and we obtain a sharp estimate for our model operator. Our results imply new -estimates for Kakeya-type maximal functions with tubes pointing along polynomial directions. Our proof technique is novel and in particular incorporates an iterated scheme of polynomial partitioning on varieties adapted to directional operators, in the vein of Guth, Guth-Katz, and Zahl.

Paper Structure

This paper contains 27 sections, 18 theorems, 164 equations.

Table of Contents

  1. Main results, motivation, background and techniques
  2. Main results
  3. Motivation
  4. Background
  5. Techniques
  6. Structure
  7. Preliminaries
  8. Notation and recurring definitions
  9. Single scale averages
  10. Decoupling of frequency annuli and localization
  11. Polynomial partition of directions
  12. Directional averages along arbitrary directions on $\mathbb S^2$
  13. Directional averages along one dimensional varieties
  14. Reduction to single annulus and polynomial partition
  15. Notions from algebraic geometry, projection varieties, and polynomial partition
  16. ...and 12 more sections

Key Result

Theorem 1

Let $n\geq 2$ and $1\leq m\leq n-1$. Let $Z_m\subseteq \mathbb{R}^n$ be a real algebraic variety of dimension $m$. Then for all $\eta>0$ there is a constant $\Theta=\Theta(Z_m,\eta)$ such that The constant $\Theta=\Theta(Z_m,\eta)$ depends on $\eta$ and on explicit algebraic properties of the variety $Z_m$. Furthermore, if $D\geq 1$ and $\mathcal{Z}^\times_{m,n}(D)$ is the class of real algebraic

Theorems & Definitions (33)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • proof
  • Lemma 2.3
  • Proposition 2.5
  • Lemma 2.6
  • proof
  • Corollary 2.7
  • proof
  • ...and 23 more