Foto: Anna Logue

Advanced Analysis (HWS 2020)

General information

--All attending student please sign up on ILIAS-- (link)
Lecturer: Dr. Georgios Psaradakis (B6 - Office C4.03)
Lecture: Mo.  10:15- 11:45, and Fri. 12:00- 13:30
Tutorial: Mo. 12:00 - 13:30
Location: WIM-Zoom-6 (see Portal2 for link and password)
Language: English
Prerequisites: Linear algebra I, Analysis I,II.

It is a master course, however, all bachelor students who have already „Analysis I and II“ are welcome to join.

The main reference will be:

  • Lieb, E. H.; Loss, M. Analysis. 2nd ed. Grad. Stud. Math. 14. Amer. Math. Soc.2001, but for measure theory we will follow the presentation in the first chapter of
  • Evans, L. C.; Gariepy, R. F. Measure theory and fine properties of functions. Stud. Adv. Math. CRC Press 1992.
  • Two books which include most of the material we plan to cover are
    • DiBenedetto, E. Real analysis. 2nd ed. Birkhäuser Adv. Texts Basler Lehr­bücher, Birkhäuser 2016,
    • Ziemer, W. P. Modern real analysis. 2nd ed. (with contributions by M. Torres), Grad. Texts in Math. 278, Springer 2017.

You can get them in the following SpringerLinks (internet connection provided by the Univ. Mannheim is required)

link.springer.com/content/pdf/10.1007%2F978-1-4939-4005-9.pdf

link.springer.com/content/pdf/10.1007%2F978-3-319-64629-9.pdf

Contents

The course splits into three parts: the first one is a crash course on measure and integration, plus a version of the Riesz representation theorem. In the second part, after learning some basic facts about convex and Lipschitz functions, we will focus on the theory of Lp spaces. With the knowledge gained from the first two parts, we will learn about: the Fourier transform in L2,the symmetric-decreasing rearrangement of functions, distributions, … . Some of the inequalities to prove: Fatou, Brunn-Minkowski, isoperimetric, Hölder, Minkowski, Hanner, Hardy-Littlewood, Polya-Szegö, one dimensional Riesz rearrangement, Young, one dimensional Hardy, Hardy-Littlewood-Sobolev, Lp-Sobolev and logarithmic Sobolev. Some facts we are going to use without proof: Hahn-Banach theorem, divergence theorem, Sard’s lemma. For the first and beginning of the second parts of the course, we will use  these lecture notes. The reference book for the course is: Lieb, E. H.; Loss, M. - Analysis (2nd ed.) Grad. Stud. Math. Vol. 14, Amer. Math. Soc. 2001

Course Materials

Office hours: Every Thursday from 9:00 to 12:00. You can also email me for an appointment.