Advanced Analysis

Dr. Georgios Psaradakis (B6 - Office C4.03)

Di.  15:30 - 17:00 in B6 -  26, A301

Do. 15:30 - 17:00 in B6 -  26, A305

Tutorial: Mo. 13:45 - 15:15 in B6 - 26, A302

Content: This course will start with some basic knowledge of real analysis, including measure and integration. Then we will go into some advanced topics in analysis, such as L^p spaces, symmetric decreasing rearrangement of functions and Riesz's rearrangement inequality, the Hardy-Littlewood-Sobolev inequality, the Fourier transform, distributions, Sobolev inequalities and the isoperimetric inequality.

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

Brief lecture notes for the first part of the course