Fourier restriction for smooth hyperbolic 2-surfaces

Abstract

We prove Fourier restriction estimates by means of the polynomial partitioning method for compact subsets of any sufficiently smooth hyperbolic hypersurface in R 3 . Our approach exploits in a crucial way the underlying hyperbolic geometry, which leads to a novel notion of strong transversality and corresponding "exceptional" sets. For the division of these exceptional sets we make crucial and perhaps surprising use of a lemma on level sets for sufficiently smooth one-variate functions from a previous article of ours.

A note on local smoothing estimates for fractional Schrödinger equations

“…In particular, the operator does not seem to have a transverse equidistribution property, which is a key ingredient in [GOW + 21b]. Even though the restriction estimate for surfaces with negative Gaussian curvature is well understood in [GO20] and [BMV20], the arguments therein do not simply rely on the properties of wave packets. Thus, it is not straightforward to the authors whether their ideas can be applied to the operator E R f .…”