Lecture: Harmonic Analysis (Winter term 2023/2024)
Lecture
Times and location
Tue 08.0009.30 in F 316a and Wed 08.0009.30 in F316a
The lecture on Wednesday, February 7, takes place at 9.4511.15 AM in F 518.
Study group on Stud.IP (Password: same as for the lecture notes)
Prerequisites:
Analysis 13.
Syllabus
Literature
 L. Grafakos  Classical Fourier Analysis (available online)
 C. D. Sogge  Fourier Integrals in Classical Analysis
 E. M. Stein  Topics in Harmonic Analysis Related to the LittlewoodPaley Theory
 E. M. Stein  Singular Integrals and Differentiability Properties of Functions (available online)
 E. M. Stein  Harmonic Analysis: RealVariable Methods, Orthogonality, and Oscillatory Integrals (available online)
 E. M. Stein and G. Weiss  Introduction to Fourier Analysis on Euclidean Spaces (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
 T. Wolff  Lectures in Harmonic Analysis (available online)
Further literature
 Crumbly notes on Fourier restriction

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
 For more details on Lorentz spaces and interpolation, see Chapter 4 in C. Bennett and R. Sharpley  Interpolation of Operators

Some basic notes on spectral theory

For more details on Stein's maximal principle, see these notes by T. Tao.

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 IIIV in Stein  Harmonic Analysis: RealVariable Methods, Orthogonality, and Oscillatory Integrals.

More on higher Riesz transforms and spherical harmonics can be found in Stein  Singular Integrals (Chapter III, § 3)

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
 Kolloquium Mathematische Physik at TU Braunschweig, organized by Professor Volker Bach
 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

November 1, 2023: Proposition 1.2.5 (7): The statement should read \((f\cdot g)^*(t_1+t_2) \leq f^*(t_1) \cdot g^*(t_2)\).

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 \(A_0=A_1=1\).

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 \(g_\epsilon\).

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 substep 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 09.4511.15 (in F 316a)

Last modified: February 5, 2024.
Back to my homepage