The lecture "Real and Harmonic Analysis" will be held at the University of Bonn in Winter 2026/2027.
Time: 14th October 2026 - 5th February 2027; always on Wednesday, 2:15 pm - 3:45 pm (Kleiner Hörsaal); Friday, 12:15 pm - 1:45 pm (Zeichensaal). (No lecture on 2nd December due to Dies Academicus; no lectures during winter break 24th December, 2026 - 6th January, 2027.)
Oral exams (30 min) will take place on 9th-11th February, 2027 and 15th - 17th March, 2027.
Office hours: Room 1.028, Thursday, 10 - 11 am. Feel free to reach out via email to schedule an appointment.
Tutorials on Wednesday, 12:15 pm - 1:45 pm by Tainara Borges and Stefanos Lappas; on Friday, 8:15 am - 9:45 am by Georg Dukanovic
Lecture notes are available on ecampus.
Topics of the lectures:
Motivation: Bilinear Cordoba--Fefferman square function estimate via Kakeya estimates; Introduction to L^p-spaces: Hölder's inequality, Minkowski's inequality, log-convexity of L^p-spaces: Riesz-Thorin
Proof of Riesz-Thorin via Hadamard three line lemma, Young's convolution inequality, Approximate identities, distribution function
Weak-L^p spaces, Convergence in measure, (Diagonal) Marcinkiewicz interpolation, Decreasing rearrangement, Lorentz spaces
Basic properties of Lorentz spaces, Bourgain's summation trick, Hardy-Littlewood maximal operator and its weak (1,1)-boundedness.
L^p-bounds for HL maximal operator, Domination of other maximal functions, Hardy-Littlewood Sobolev inequality. Basic properties of the Schwartz functions and the Fourier transform
The Fourier transform on L^1 + L^2 and on the space of tempered distributions, weak^*-continuity, examples of Fourier transforms in the sense of tempered distributions
Characterization of translation-invariant operators between L^p-spaces as convolution with tempered distribution, Identification of translation-invariant bounded operators L^1-L^1 with finite Borel measures
Identification of translation-invariant bounded operators L^2-L^2 with bounded Fourier multipliers, non-stationary phase estimates; Fourier series: Dirichlet and F\'ejer kernel, density of trigonometric polynomials in L^p(\T^d)