Weighted restriction estimates and application to Falconer distance set problem

TL;DR

This paper develops weighted Fourier restriction estimates using polynomial partitioning and refined Strichartz estimates, leading to improved decay rates of fractal measures' Fourier transforms and advancing the Falconer distance set conjecture in higher dimensions.

Contribution

The paper introduces new weighted restriction estimates and applies them to improve results related to the Falconer distance set conjecture in multiple dimensions.

Findings

Improved spherical average decay rates of Fourier transforms of fractal measures.

Enhanced results for the Falconer distance set conjecture in three and higher dimensions.

Application of polynomial partitioning and refined Strichartz estimates in restriction theory.

Abstract

We prove some weighted Fourier restriction estimates using polynomial partitioning and refined Strichartz estimates. As application we obtain improved spherical average decay rates of the Fourier transform of fractal measures, and therefore improve the results for the Falconer distance set conjecture in three and higher dimensions.