$L^2$ affine Fourier restriction theorems for smooth surfaces in $\mathbb{R}^3$

TL;DR

This paper establishes sharp $L^2$ Fourier restriction inequalities for smooth surfaces in three-dimensional space, using decoupling techniques, with uniform bounds for polynomial graphs of bounded degree.

Contribution

It introduces a decoupling theorem for affine surface measures on smooth surfaces, providing uniform restriction bounds applicable to polynomial graphs of bounded degree.

Findings

Sharp $L^2$ restriction inequalities proven

Results hold uniformly for polynomial graphs of bounded degree

Decoupling theorem developed as the main tool

Abstract

We prove sharp $L^2$ Fourier restriction inequalities for compact, smooth surfaces in $\mathbb{R}^3$ equipped with the affine surface measure or a power thereof. The results are valid for all smooth surfaces and the bounds are uniform for all surfaces defined by the graph of polynomials of degrees up to $d$ with bounded coefficients. The primary tool is a decoupling theorem for these surfaces.