Weighted restriction estimates using polynomial partitioning

TL;DR

This paper employs polynomial partitioning to establish new weighted Fourier restriction estimates in three dimensions, leading to improved unweighted restriction results for smooth surfaces and insights into decay properties of Fourier transforms of measures.

Contribution

The paper introduces a novel application of polynomial partitioning to weighted restriction estimates in bb R^3, extending the range of exponents and deriving new decay results for Fourier transforms of measures.

Findings

Established weighted restriction estimates for exponents between 3 and 3.25.

Proved new global restriction estimates for smooth surfaces with positive curvature.

Derived decay results for Fourier transforms of measures with finite imensional energies.

Abstract

We use the polynomial partitioning method of Guth to prove weighted Fourier restriction estimates in $\Bbb R^3$ with exponents $p$ that range between $3$ and $3.25$, depending on the weight. As a corollary to our main theorem, we obtain new (non-weighted) local and global restriction estimates for compact $C^\infty$ surfaces $S \subset \Bbb R^3$ with strictly positive second fundamental form. For example, we establish the global restriction estimate $\| Ef \|_{L^p(\Bbb R^3)} \leq C \, \| f \|_{L^q(S)}$ in the full conjectured range of exponents for $p > 3.25$ (up to the sharp line), and the global restriction estimate $\| Ef \|_{L^p(\Omega)} \leq C \, \| f \|_{L^2(S)}$ for $p>3$ and certain sets $\Omega \subset \Bbb R^3$ of infinite Lebesgue measure. As a corollary to our main theorem, we also obtain new results on the decay of spherical means of Fourier transforms of positive compactly supported measures on $\Bbb R^3$ with finite $\alpha$-dimensional energies.