Lecture: Harmonic Analysis (Winter term 2023/2024)

Lecture
Times and location
Tue 08.00-09.30 in F 316a and Wed 08.00-09.30 in F316a
The lecture on Wednesday, February 7, takes place at 9.45-11.15 AM in F 518.
Study group on Stud.IP (Password: same as for the lecture notes)

Prerequisites: Analysis 1-3.

Syllabus

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)

1. Crumbly notes on Fourier restriction
2. 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
3. For more details on Lorentz spaces and interpolation, see Chapter 4 in C. Bennett and R. Sharpley - Interpolation of Operators
4. Some basic notes on spectral theory
5. For more details on Stein's maximal principle, see these notes by T. Tao.
6. 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.
7. More on higher Riesz transforms and spherical harmonics can be found in Stein - Singular Integrals (Chapter III, § 3)
8. See this video for an explanation of Kakeya's needle problem by Charles Fefferman. Here are also some notes and slides. (Open the slides on your pdf viewer to play the videos.)

Local seminars on mathematical physics

1. Kolloquium Mathematische Physik at TU Braunschweig, organized by Professor Volker Bach
2. Online GAuS seminar on topics in analysis, partial differential equations, and mathematical physics

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

Corrigenda

1. November 1, 2023: Proposition 1.2.5 (7): The statement should read \((f\cdot g)^*(t_1+t_2) \leq f^*(t_1) \cdot g^*(t_2)\).
2. November 15, 2023: In the proof of Marcinkiewicz' interpolation theorem (Theorem 1.3.4), we can assume without loss of generality that \(A_0=A_1=1\) and, therefore, that \(A_\theta=1\). To see this, we use the homogeneity of \(T\), \(\mu\), and \(\nu\) and multiply \(T\) with a constant \(C_1\), \(\mu\) with a constant \(C_2\), \(\nu\) with a constant \(C_3\), and \(A_\theta\) with \(C_1C_2^{-1/p_\theta}C_3^{1/q_\theta}\). Then the claimed restricted \((p_\theta,q_\theta)\) estimate remains untouched but we have enough freedom to pick \(C_1,C_2,C_3\) to normalize \(A_0=A_1=1\).
3. November 15, 2023: Lemma 1.3.5: In the statement of this lemma, we should merely assume that there is a \(\delta>0\) such that \(|f(z)|\) satisfies the claimed double exponential growth. See Theorem 14 in these Complex Analysis notes of T. Tao for a choice of the counter functions \(g_\epsilon\).
4. November 21, 2023: Propositions 1.3.2 and 1.3.3: A function \(f\) is called simple if there are \(a_j\in\mathbb{C}\), \(E_j\subseteq\mathbb{R}^d\), and \(M\in\mathbb{N}\) such that \(f(x)=\sum_{j=1}^M a_j\mathbf{1}_{E_j}\). A simple sub-step function \(f\) of height \(H\) and width \(W\) is a simple function obeying \(|f|\lesssim H\mathbf{1}_E\) with \(|E|\lesssim W\).

Exercises
Tue 09.45-11.15 (in F 316a)