Lecture: Harmonic Analysis (SS 2020)

Wed 16.45-18.15 (in F 513) and Thu 16.45-18.15 (in ????), starting on April 22, 2020
There will be no lectures in the week of June 1.

In view of the decision of the state secretaries of the department of sciences, the beginning of 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. L. Grafakos - Classical Fourier Analysis (available online)
  2. C.D. Sogge - Fourier Integrals in Classical Analysis
  3. E. M. Stein - Topics in Harmonic Analysis Related to the Littlewood-Paley Theory
  4. E. M. Stein - Singular Integrals and Differentiability Properties of Functions (available online)
  5. E. M. Stein - Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals (available online)
  6. E. M. Stein and G. Weiss - Introduction to Fourier Analysis on Euclidean Spaces (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. T. Wolff - Lectures in Harmonic Analysis (available online)
Further literature will be announced here.
  1. More on decreasing rearrangements and their applications (in isoperimetric inequalities) can be found, e.g., here: R. D. Benguria - Isoperimetric inequalities for eigenvalues of the Laplacian
  2. For more details on Lorentz spaces and interpolation, see Chapter 4 in C. Bennett and R. Sharpley - Interpolation of Operators
  3. Some basic notes on spectral theory
  4. More on singular integrals on weighted L^p, the first Hardy space H^1 (instead of L^1) and the relation between H^1 and BMO can be found in Chapters III-V in Stein - Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals.
  5. More on higher Riesz transforms and spherical harmonics can be found in Stein - Singular Integrals (Chapter III, § 3)


Course material
Lecture 1 (Lorentz spaces and interpolation)
Lecture 2 (Maximal functions and covering lemmas) (preliminary notes)
Lecture 3 (Singular integrals) (preliminary notes)
Lecture 4 (Topics in Fourier analysis) (preliminary notes)

Thu 8.00-9.30 (in F 315)

Exercise sheets
Sheet 1
Sheet 2
Modified the hints in exercises 2.3 and the assumptions in exercise 2.4. Hint to exercise 2.3
Sheet 3
Sheet 4
Sheet 5
To be discussed on June 11 (although exercise 5.1 could already be discussed on May 28)
Sheet 6
Sheet 7
Replaced exercise 7.3
Sheet 8
Sheet 9
Corrected a typo in exercise 9.1
Sheet 10
Sheet 11


Last modified: July 23, 2020.

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