A restriction estimate using polynomial partitioning

TL;DR

This paper establishes new restriction estimates for smooth surfaces in three-dimensional space using polynomial partitioning, improving understanding of Fourier extension operators for certain p-values.

Contribution

It introduces a novel application of polynomial partitioning to restriction estimates for surfaces with positive curvature in $\,\,\mathbb{R}^3$.

Findings

Proves restriction estimates for p > 3.25

Uses polynomial partitioning techniques from incidence geometry

Provides bounds for extension operators on smooth surfaces

Abstract

If $S$ is a smooth compact surface in $\mathbb{R}^3$ with strictly positive second fundamental form, and $E_S$ is the corresponding extension operator, then we prove that for all $p > 3.25$, $\| E_S f\|_{L^p(\mathbb{R}^3)} \le C(p,S) \| f \|_{L^\infty(S)}$. The proof uses polynomial partitioning arguments from incidence geometry.