Maximal directional operators along algebraic varieties

TL;DR

This paper determines the precise growth rate of the $L^2$-norm of maximal directional averaging operators along algebraic varieties, extending previous results and improving bounds for directions on the sphere, with applications to Kakeya-type maximal functions.

Contribution

It introduces a novel proof technique using polynomial partitioning on varieties, extending maximal operator bounds to arbitrary-dimensional algebraic varieties.

Findings

Established sharp $L^2$ bounds for maximal directional operators along algebraic varieties.

Improved bounds for directions on the 2-sphere by a factor of $ oot{2}\log N$.

Derived new $L^2$ estimates for Kakeya-type maximal functions.

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 C\'ordoba, 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.