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

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.

arXiv commentAccepted by Indiana University Mathematics Journal in Nov 2024; incorporated referee's comments, and updated references. This version is final