A restriction estimate in $\mathbb{R}^3$ using brooms

TL;DR

This paper establishes a new restriction estimate for the paraboloid in three-dimensional space, improving bounds for the extension operator using polynomial partitioning and geometric wave packet analysis.

Contribution

It introduces a restriction estimate for the paraboloid in D using novel combination of Wolff's two ends argument and polynomial partitioning techniques.

Findings

Proves restriction estimate for p > 3 + 3/13 in D

Combines wave packet geometry with polynomial partitioning methods

Identifies geometric structures in wave packets

Abstract

If $f$ is a function supported on the truncated paraboloid in $\mathbb{R}^3$ and $E$ is the corresponding extension operator, then we prove that for all $p> 3+ 3/13$, $\|Ef\|_{L^p(\mathbb{R}^3)}\leq C \|f\|_{L^{\infty}}$. The proof combines Wolff's two ends argument with polynomial partitioning techniques. We also observe some geometric structures in wave packets.