$L^2$ bounds for a Kakeya type maximal operator in $\R^3$

TL;DR

This paper establishes near-optimal $L^2$ bounds for a Kakeya-type maximal operator in three dimensions, depending on the number and separation of directions, with implications for harmonic analysis.

Contribution

It provides new bounds for a maximal operator in $ ^3$ with arbitrary and separated directions, improving previous estimates and approaching optimality.

Findings

Bound of $N^{1/4}{ m log} N$ for arbitrary directions

Improved bound of $N^{1/4} oot{ m log} N$ for separated directions

Bounds are nearly optimal apart from logarithmic factors

Abstract

We prove that the maximal operator obtained by taking averages at scale 1 along $N$ arbitrary directions on the sphere, is bounded in $L^2(\R^3)$ by $N^{1/4}{\log N}$. When the directions are $N^{-1/2}$ separated, we improve the bound to $N^{1/4}\sqrt{\log N}$. Apart from the logarithmic terms these bounds are optimal.