A family of fractal Fourier restriction estimates with implications on the Kakeya problem

TL;DR

This paper introduces a family of fractal Fourier restriction estimates that extend previous results and have significant implications for the Kakeya problem, including new sharp estimates in two dimensions.

Contribution

It establishes a continuum of fractal restriction estimates, each implying sharp Kakeya results related to polynomial Wolff axioms, extending prior work by Du and Zhang.

Findings

The Du-Zhang estimate is the endpoint of a family of fractal restriction estimates.

All estimates in the family hold in  b0.

Each estimate (except the original) implies a sharp Kakeya result.

Abstract

In a recent paper [Ann. of Math. 189 (2019), 837--861], Du and Zhang proved a fractal Fourier restriction estimate and used it to establish the sharp $L^2$ estimate on the Schr\"{o}dinger maximal function in $\Bbb R^n$, $n \geq 2$. In this paper, we show that the Du-Zhang estimate is the endpoint of a family of fractal restriction estimates such that each member of the family (other than the original) implies a sharp Kakeya result in $\Bbb R^n$ that is closely related to the polynomial Wolff axioms. We also prove that all the estimates of our family are true in $\Bbb R^2$.