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On a Sharp Fourier Extension Inequality on the Circle with Lacunary Spectrum

Felipe Gonçalves, João Paulo Ferreira

TL;DR

The paper proves a sharp Tomas–Stein type Fourier extension inequality on the circle for functions whose spectrum is lacunary with ratio $\lambda_{n+1}/\lambda_n>3$, with equality only when $\widehat{f}(k)=0$ for $k\neq0$. The approach combines an explicit representation of $\|\widehat{f\sigma}\|_{L^6(\mathbb{R}^2)}^6$ via Bessel functions and a decomposition of $A^3$ into $A_{\text{P(3)}}^3$ and $A_E^3$, controlled through a ratio function $F$ and a parameter $b>1$; a tunable $\varepsilon_D$ handles cross-term contributions. A key step is the precise description of $A_E^3$ (Lemma P3exceptions) and lower bounds on $F$ (Lemma BoundsforF), including numerically assisted estimates for small indices using PARI-GP. These ingredients yield the sharp inequality for lacunary constants $q\ge3$, with equality only for the trivial spectrum, extending prior results for band-limited spectra and supporting conjectured extremizer structures in 2D Fourier restriction on conic sections.

Abstract

We prove a sharp Fourier extension inequality on the circle for the Tomas-Stein exponent for functions whose spectrum $\{\pm λ_n\}$ satisfies $λ_{n+1}>3 λ_{n}$.

On a Sharp Fourier Extension Inequality on the Circle with Lacunary Spectrum

TL;DR

The paper proves a sharp Tomas–Stein type Fourier extension inequality on the circle for functions whose spectrum is lacunary with ratio , with equality only when for . The approach combines an explicit representation of via Bessel functions and a decomposition of into and , controlled through a ratio function and a parameter ; a tunable handles cross-term contributions. A key step is the precise description of (Lemma P3exceptions) and lower bounds on (Lemma BoundsforF), including numerically assisted estimates for small indices using PARI-GP. These ingredients yield the sharp inequality for lacunary constants , with equality only for the trivial spectrum, extending prior results for band-limited spectra and supporting conjectured extremizer structures in 2D Fourier restriction on conic sections.

Abstract

We prove a sharp Fourier extension inequality on the circle for the Tomas-Stein exponent for functions whose spectrum satisfies .
Paper Structure (5 sections, 4 theorems, 31 equations)

This paper contains 5 sections, 4 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Motivation, Notation and Strategy
  3. Further Difficulties
  4. Preliminary Results
  5. Proof of Main Result

Key Result

Theorem 1

Let $f\in L^2(\mathbb{S}^1)$ be such that $\text{spec}(f)\subset A_{\lambda,q}:=\{\pm \lambda_n;n\geq 0\}$, where $\{\lambda_n\}$ is a lacunary sequence of non-negative integers satisfying $\lambda_{n+1}/\lambda_{n}>3$, and $\lambda_{0}:=0$. Then and equality is attained if and only if $\widehat{f}(k)=0$ for every $k\neq 0$.

Theorems & Definitions (12)

  • Conjecture 1: Case $d=2$
  • Theorem 1
  • Definition 2
  • Definition 3
  • Lemma 4
  • Example 5
  • proof : Proof of Lemma \ref{['P3exceptions']}
  • Remark 6
  • Lemma 7
  • Lemma 8
  • ...and 2 more