Lecture: Fourier Restriction and Applications (SS 2025)



Lecture
Tue 16.45-18.15 in PK 4.4 and Thu 09.45-11.15 in UP 2.316a, starting on April 8, 2025


Prerequisites
Analysis 1-3. A nodding acquaintance with functional analysis, distribution theory and some familiarity with the Fourier transform are helpful but not necessary.

Literature
  1. C. Demeter - Fourier Restriction, Decoupling, and Applications
  2. L. Grafakos - Classical Fourier Analysis (available online)
  3. P. Mattila - Fourier Analysis and Hausdorff Dimension
  4. C.D. Sogge - Fourier Integrals in Classical Analysis
  5. C.D. Sogge - Hangzhou Lectures on Eigenfunctions of the Laplacian
  6. E. M. Stein - Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals (available online)
  7. T. Tao - Notes on Fourier analysis
  8. T. Tao - Topics in real analysis: restriction theorems, Besicovitch sets, and applications to PDE
  9. T. Tao - Recent progress on the restriction conjecture
  10. R. Vershynin - High-Dimensional Probability
  11. T. Wolff - Lectures in Harmonic Analysis (available online)
  12. D. R. Yafaev - Mathematical Scattering Theory. Analytic Theory
Further literature will be announced here.
  1. Some connections between Fourier analysis and number theory reviewed by Larry Guth: ICM 2022 lecture and ICM 2022 slides
  2. Crumbly notes
  3. More on the locally constant lemma, wave packet decomposition
  4. More precise estimates for Bessel functions can be found here, in Barcelo-Ruiz-Vega, and in Cordoba
  5. Sharp Tomas-Stein Fourier restriction with vanishing curvatures in three dimensions was carried out by Ikromov and Müller
  6. For more general versions and an optimal extension of the Tomas-Stein theorem to Lorentz spaces, see Bak and Seeger
  7. References for randomized restriction theorems: dual to Sudakov by Pajor-Tomczak--Jaegermann and Bourgain-Lindenstrauss-Milman. Random trace lemma by Bourgain and random Tomas-Stein by Bourgain


Syllabus

Course material
Lecture 1 (Basics in harmonic analysis) (preliminary notes)
Lecture 2 (Basics in Fourier analysis) (preliminary notes)
Lecture 3 (Oscillatory integrals) (preliminary notes)
Lecture 4 (Tomas-Stein and applications) (preliminary notes)
Lecture 5 (Two-dimensional restriction theorems) (preliminary notes)
Lecture 6 (Relation between restriction and Kakeya conjectures) (preliminary notes)

Exercises
Tue 09.45-11.15 every second week. The first class is on April 22.

Exercise sheets
Remarks
Sheet 1
To be discussed on April 22. We will discuss Ex 1.3 and 1.5 on May 13.
Sheet 2
To be discussed on May 13.
Sheet 3
To be discussed on May 27. Addendum to exercise 3.3.
Sheet 4
To be discussed on June 17.
Sheet 5
To be discussed on July 1
Sheet 6
Not to be discussed in class. Questions welcome.


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Last modified: July 03, 2025.



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