Lecture: Fourier Restriction and Applications (SS 2021)

Wed 16.45-18.15 andcThu 11.30-13.00, starting on April 14, 2021

This lecture will take place online. For this reason, students who are planning to participate in this course are advised to write an e-mail to k DOT merz AT tu-bs DOT de.

Analysis 1-3. A nodding acquaintance with functional analysis, distribution theory and some familiarity with the Fourier transform are helpful.

  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. Crumbly notes
  2. More on the locally constant lemma, wave packet decomposition
  3. More precise estimates for Bessel functions can be found here, in Barcelo-Ruiz-Vega, and in Cordoba
  4. Sharp Tomas-Stein Fourier restriction with vanishing curvatures in three dimensions was carried out by Ikromov and Müller
  5. For more general versions and an optimal extension of the Tomas-Stein theorem to Lorentz spaces, see Bak and Seeger
  6. 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


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

Wed 16.45-18.15 every second week. The first class is on April 28.

Exercise sheets
Sheet 1
To be discussed on April 28. Corrected an assumption in Ex 1.4.
Sheet 2
Sheet 3
Extension of exercises 3.3 and 3.4 to more general manifolds.
Sheet 4
Corrected the statement in exercise 4.1
Sheet 5
To be discussed earliest on July 7 (otherwise July 14)


Last modified: March 22, 2021.

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