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.

- C. Demeter - Fourier Restriction, Decoupling, and Applications
- L. Grafakos - Classical Fourier Analysis (available online)
- P. Mattila - Fourier Analysis and Hausdorff Dimension
- C.D. Sogge - Fourier Integrals in Classical Analysis
- C.D. Sogge - Hangzhou Lectures on Eigenfunctions of the Laplacian
- E. M. Stein - Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals (available online)
- T. Tao - Notes on Fourier analysis
- T. Tao - Topics in real analysis: restriction theorems, Besicovitch sets, and applications to PDE
- T. Tao - Recent progress on the restriction conjecture
- R. Vershynin - High-Dimensional Probability
- T. Wolff - Lectures in Harmonic Analysis (available online)
- D. R. Yafaev - Mathematical Scattering Theory. Analytic Theory

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

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 |
Remarks |

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Last modified: March 22, 2021.

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