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$L^2$ restriction estimates from the Fourier spectrum

Marc Carnovale, Jonathan M. Fraser, Ana E. de Orellana

TL;DR

This work introduces the Fourier spectrum as a refined descriptive tool for restriction problems, providing a continuum of $L^{q'}\to L^2$ restriction estimates that interpolate between classical Frostman-based and Sobolev-based perspectives. By optimizing over the spectrum parameter $\theta$, the authors derive sharper ranges than the Stein--Tomas theorem for a variety of measures, including the cone, the moment curve, and fractal constructions, and they also establish a partial converse describing when restriction fails. The results are extended to Lorentz spaces and endpoint cases, yielding sharper, near-optimal bounds and new endpoint phenomena. The paper demonstrates concrete advantages through detailed applications and computations of the Fourier spectrum for key measures, highlighting phase transitions and the interplay between Frostman, Fourier, and Sobolev dimensions in fractal geometry.

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 Łaba. We illustrate our results with several examples, including the surface measure on the cone, the moment curve, and several fractal measures.

$L^2$ restriction estimates from the Fourier spectrum

TL;DR

This work introduces the Fourier spectrum as a refined descriptive tool for restriction problems, providing a continuum of restriction estimates that interpolate between classical Frostman-based and Sobolev-based perspectives. By optimizing over the spectrum parameter , the authors derive sharper ranges than the Stein--Tomas theorem for a variety of measures, including the cone, the moment curve, and fractal constructions, and they also establish a partial converse describing when restriction fails. The results are extended to Lorentz spaces and endpoint cases, yielding sharper, near-optimal bounds and new endpoint phenomena. The paper demonstrates concrete advantages through detailed applications and computations of the Fourier spectrum for key measures, highlighting phase transitions and the interplay between Frostman, Fourier, and Sobolev dimensions in fractal geometry.

Abstract

The Stein--Tomas restriction theorem is an important result in Fourier restriction theory. It gives a range of for which 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 restriction theorem which often outperforms the Stein--Tomas result. We also provide results in the other direction by giving a range of in terms of the Fourier spectrum for which restriction estimates fail, generalising an observation of Hambrook and Łaba. We illustrate our results with several examples, including the surface measure on the cone, the moment curve, and several fractal measures.

Paper Structure

This paper contains 24 sections, 13 theorems, 161 equations, 6 figures.

Table of Contents

  1. Introduction
  2. The restriction problem
  3. Stein--Tomas restriction
  4. New Stein--Tomas type results using the Fourier spectrum
  5. Preliminaries
  6. Frostman, Hausdorff and Fourier dimensions
  7. The Fourier spectrum
  8. Main results
  9. $L^2$ restriction estimates from the Fourier spectrum
  10. Proof of Theorem \ref{['thm:mainthm']}
  11. A partial converse from the Fourier spectrum
  12. Restriction estimates for Lorentz spaces and the endpoint
  13. Lorentz spaces and interpolation
  14. Restriction estimates for Lorentz spaces
  15. Proof of Theorem \ref{['thm:endpoint']}
  16. ...and 9 more sections

Key Result

Theorem 3.1

Let $\mu$ be a non-zero, finite, compactly supported, Borel measure on $\mathbb{R}^d$, with $\dim_{\textup{Fr}}\mu=\alpha$ for some $0<\alpha< d$. If then for all $f\in L^2(\mu)$, Equivalently, if then for all $f\in L^{q'}(\mathbb{R}^d)$,

Figures (6)

  • Figure 1: In order to improve the Stein--Tomas range for the restriction problem, we need the Fourier spectrum of $\mu$ to intersect the shaded region, i.e. for some $\theta\in[0,1]$ we need the point ($\theta, \dim^\theta_{\mathrm{F}} \mu)$ to lie in the shaded region. Top left: when $\dim_{\mathrm{S}}\mu\geqslant\dim_{\textup{Fr}}\mu+\frac{\dim_{\mathrm{F}}\mu}{2}$ and $\dim_{\textup{Fr}}\mu>\dim_{\mathrm{F}}\mu$. Top right: when $\dim_{\mathrm{S}}\mu>\dim_{\textup{Fr}}\mu + \frac{\dim_{\mathrm{F}}\mu}{2}$ and $\dim_{\textup{Fr}}\mu<\dim_{\mathrm{F}}\mu$. Bottom left: when $\dim_{\mathrm{S}}\mu<\dim_{\textup{Fr}}\mu+\frac{\dim_{\mathrm{F}}\mu}{2}$ and $\dim_{\textup{Fr}}\mu < \dim_{\mathrm{F}}\mu$. Bottom right: when $\dim_{\mathrm{S}}\mu<\dim_{\textup{Fr}}\mu+\frac{\dim_{\mathrm{F}}\mu}{2}$ and $\dim_{\textup{Fr}}\mu > \dim_{\mathrm{F}}\mu$. The dashed lines are $\theta(\dim_{\textup{Fr}}\mu - \tfrac{\dim_{\mathrm{F}}\mu}{2}) + \dim_{\mathrm{F}}\mu$ and $d\theta$, and the solid line is $\theta(\dim_{\mathrm{S}}\mu - \dim_{\mathrm{F}}\mu) + \dim_{\mathrm{F}}\mu$, which by concavity is always a lower bound for the Fourier spectrum of $\mu$.
  • Figure 2: The Fourier spectrum of $\nu_{d-1}$ on the cone $C^{d-1}$ for $d = 3, \ldots, 6$; see Proposition \ref{['thm:coneSpectrum']}.
  • Figure 3: Bounds for the range of $q$ for the restriction estimate \ref{['eq:extension']} to hold for the cone in $\mathbb{R}^d$. The dashed lines are the Stein--Tomas upper bound and the Hambrook--Łaba lower bound, the dotted line is the sharp result, and the solid lines are our upper and lower bounds for the threshold. These plots should be understood as only applying to integer points in the domain, but we included the full curve for aesthetic reasons.
  • Figure 4: The Fourier spectrum of the arclength measure on the moment curve in $\mathbb{R}^8$; see Proposition \ref{['prop:moment']}. There are 6 phase transitions and the Fourier dimension is 1/4.
  • Figure 5: Bounds for the range of $q$ for the extension estimate \ref{['eq:extension']} to hold for the moment curve in $\mathbb{R}^d$. The dashed lines are the Stein--Tomas upper bound and the Hambrook--Łaba lower bound, the dotted line is the sharp result, and the solid lines are our upper and lower bounds for the threshold. These plots should be understood as only applying to integer points in the domain, but we included the full curve for aesthetic reasons.
  • ...and 1 more figures

Theorems & Definitions (17)

  • Theorem 3.1
  • Theorem 3.2
  • Lemma 3.3
  • proof
  • Corollary 3.4
  • Corollary 3.5
  • Theorem 3.6
  • proof
  • Corollary 3.7
  • Lemma 4.1: Bourgain's interpolation trick
  • ...and 7 more