Scale invariant Fourier restriction to a hyperbolic surface

TL;DR

This paper establishes the first scale-invariant Fourier restriction estimates for a hyperbolic surface with mixed curvatures, advancing the understanding of extension estimates beyond classical ranges.

Contribution

It sharpens the bilinear to linear deduction for hyperbolic paraboloids, achieving the first scale-invariant restriction estimates beyond the Stein--Tomas range.

Findings

First scale-invariant restriction estimates for hyperbolic surfaces

Extension of bilinear to linear estimates in this setting

Results applicable to hypersurfaces with mixed principal curvatures

Abstract

This result sharpens the bilinear to linear deduction of Lee and Vargas for extension estimates on the hyperbolic paraboloid in $\mathbb R^3$ to the sharp line, leading to the first scale-invariant restriction estimates, beyond the Stein--Tomas range, for a hypersurface on which the principal curvatures have different signs.