$L^2$ restriction estimates from the Fourier spectrum

TL;DR

This paper extends the Stein--Tomas restriction theorem by using the Fourier spectrum to interpolate between dimensions, resulting in improved restriction estimates for various measures, including fractals.

Contribution

It introduces a new framework using the Fourier spectrum to derive a continuum of restriction estimates, surpassing classical results in many cases.

Findings

New restriction estimates often outperform Stein--Tomas bounds.

Range of q for restriction failure is characterized via Fourier spectrum.

Applications include fractal measures and classical geometric surfaces.

Abstract

The Stein--Tomas restriction theorem is an important result in Fourier restriction theory. It gives a range of $q$ for which $L^q\to L^2$ restriction estimates hold for a given measure, in terms of the Fourier and Frostman dimensions of the measure. We generalise this result by using the Fourier spectrum; a family of dimensions that interpolate between the Fourier and Sobolev dimensions for measures. This gives us a continuum of Stein--Tomas type estimates, and optimising over this continuum gives a new $L^q\to L^2$ restriction theorem which often outperforms the Stein--Tomas result. We also provide results in the other direction by giving a range of $q$ in terms of the Fourier spectrum for which $L^q\to L^2$ restriction estimates fail, generalising an observation of Hambrook and {\L}aba. We illustrate our results with several examples, including the surface measure on the cone, the moment curve, and several fractal measures.