Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Some geometric and spectral aspects of restriction problems

Hajer Bahouri, Veronique Fischer

Abstract

This texts commemorates the memory of Haim Brezis and explores some aspects of the restriction problem, particularly its connections to spectral and geometric analysis. Our choice of subject is motivated by Brezis' significant contributions to various domains related to this problem, including harmonic analysis, partial differential equations, spectral theory, representation theory, number theory, and many others.

Some geometric and spectral aspects of restriction problems

Abstract

This texts commemorates the memory of Haim Brezis and explores some aspects of the restriction problem, particularly its connections to spectral and geometric analysis. Our choice of subject is motivated by Brezis' significant contributions to various domains related to this problem, including harmonic analysis, partial differential equations, spectral theory, representation theory, number theory, and many others.

Paper Structure

This paper contains 53 sections, 14 theorems, 166 equations.

Table of Contents

  1. Introduction
  2. The first $L^2$-restriction theorem: on the sphere
  3. Statement
  4. Proof of Theorem \ref{['thm_1rest']}
  5. Tomas' improvement
  6. Tomas-Stein Restriction theorem on hyper-surfaces
  7. The induced measure on a hyper-surface
  8. Definition
  9. Estimates for the Fourier transform of the induced measure
  10. Measure theoretic description
  11. Statement of the Tomas-Stein Restriction Theorem
  12. Spectral viewpoint
  13. Spectral description of the operator $\mathcal{P}$
  14. The operator $\mathcal{P}$
  15. Dilations properties
  16. ...and 38 more sections

Key Result

Theorem 2.1

Let $n\geq 2$ and $p\in [1,4n /(3n+1))$. The estimates in eq:estimesphere hold for any $f\in \mathcal{S}(\mathbb R^n)$.

Theorems & Definitions (36)

  • Theorem 2.1: First restriction theorem
  • proof : Proof of Theorem \ref{['thm_1rest']}
  • Theorem 2.2: Tomas
  • proof : Sketch of the proof of Theorem \ref{['thm_Tomas']} for $p\in [1,p_{TS})$
  • Example 1
  • Example 2
  • Example 3
  • Theorem 3.1: stein
  • proof : Sketch of the proof of Theorem \ref{['thm_estwhsigma']}
  • Theorem 3.2: Tomasstein
  • ...and 26 more