Maximal directional operators along algebraic varieties

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.

34 pages, final version, incorporates the comments of the anonymous referees; to appear in Amer. J. Math

References in corpus (4)

• Square functions for bi-Lipschitz maps and directional operators
• A cone restriction estimate using polynomial partitioning
• A sharp estimate for the Hilbert transform along finite order lacunary sets of directions
• On the maximal directional Hilbert transform in three dimensions

Cited by in corpus (5)

• Directional square functions
• $L^2$ Bounds for a maximal directional Hilbert transform
• Maximal subspace averages
• Almost-orthogonality principles for certain directional maximal functions
• Discrete Analogues in Harmonic Analysis: Directional Maximal Functions in ${\mathbb{Z}}^2$