Improved weighted restriction estimates in $\Bbb R^3$

TL;DR

This paper advances weighted restriction estimates for the paraboloid in three dimensions, achieving optimal exponent ranges and enhancing understanding of Fourier transform decay and geometric measure theory implications.

Contribution

It improves known weighted restriction estimates for the paraboloid in R^3, reaching the full range of exponents up to the sharp line, based on polynomial partitioning techniques.

Findings

Enhanced restriction estimates for the paraboloid in R^3.

Achieved the full possible range of p,q exponents up to the sharp line.

Results have implications for Fourier decay and Falconer's distance set conjecture.

Abstract

Suppose $0 < \alpha \leq n$, $H: \Bbb R^n \to [0,1]$ is a Lebesgue measurable function, and $A_\alpha(H)$ is the infimum of all numbers $C$ for which the inequality $\int_B H(x) dx \leq C R^\alpha$ holds for all balls $B \subset \Bbb R^n$ of radius $R \geq 1$. After Guth introduced polynomial partitioning to Fourier restriction theory, weighted restriction estimates of the form $\| Ef \|_{L^p(B,Hdx)} \leq C R^\epsilon A_\alpha(H)^{1/p} \| f \|_{L^q(\sigma)}$ have been studied and proved in several papers, leading to new results about the decay properties of spherical means of Fourier transforms of measures and, in some cases, to progress on Falconer's distance set conjecture in geometric measure theory. This paper improves on the known estimates when $E$ is the extension operator associated with the unit paraboloid ${\mathcal P} \subset \Bbb R^3$, reaching the full possible range of $p,q$ exponents (up to the sharp line) for $p \geq 3 + (\alpha-2)/(\alpha+1)$ and $2 < \alpha \leq 3$.