On Wolff's $L^{\frac{5}{2}}-$Kakeya maximal inequality in $\R^3$

TL;DR

This paper provides a new proof of Wolff's $L^{5/2}$ bound for the Kakeya maximal function in three dimensions, using a geometric approach inspired by Sogge's strategy, avoiding induction on scales.

Contribution

The authors offer a novel proof technique for Wolff's Kakeya maximal inequality, connecting it with Sogge's approach and revealing new geometric insights.

Findings

Reproves Wolff's $L^{5/2}$ bound without induction on scales

Introduces a geometric method inspired by Sogge's strategy

Highlights new geometric observations related to Kakeya sets

Abstract

We reprove Wolff's $L^{\frac{5}2}-$ bound for the $\R^3-$Kakeya maximal function without appealing to the argument of induction on scales. The main ingredient in our proof is an adaptation of Sogge's strategy used in the work on Nikodym-type sets in curved spaces. Although the equivalence between these two type maximal functions is well known, our proof may shed light on some new geometric observations which is interesting in its own right.