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Sharp Fourier Extension on the Circle Under Arithmetic Constraints

Valentina Ciccone, Felipe Gonçalves

Abstract

We establish a sharp adjoint Fourier restriction inequality for the end-point Tomas-Stein restriction theorem on the circle under a certain arithmetic constraint on the support set of the Fourier coefficients of the given function. Such arithmetic constraint is a generalization of a $B_3$-set.

Sharp Fourier Extension on the Circle Under Arithmetic Constraints

Abstract

We establish a sharp adjoint Fourier restriction inequality for the end-point Tomas-Stein restriction theorem on the circle under a certain arithmetic constraint on the support set of the Fourier coefficients of the given function. Such arithmetic constraint is a generalization of a -set.
Paper Structure (7 sections, 4 theorems, 65 equations)

This paper contains 7 sections, 4 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. A Generalization of $B_h$-sets
  4. Examples of $\mathrm{P}(h)$-sets
  5. Estimates for certain integrals of Bessel functions
  6. Proof of the main result
  7. A further example of application

Key Result

Theorem 1

Let $f\in L^2(\mathbb{S}^1)$ be such that its spectrum is a $\mathrm{P}(3)$-set. Then and equality is attained if and only if $f$ is constant.

Theorems & Definitions (11)

  • Definition 1
  • Theorem 1
  • Definition 2: Property $\mathrm{P}(h)$
  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Lemma 2
  • Lemma 3
  • proof
  • ...and 1 more