# Advaned analysis

Dr. Georgios Psaradakis

Di. 8:30 - 10:00 in A5,6 - Hilbertraum (C116)

Do. 12:00 - 13:30 in A5,6 - Hilbertraum (C116)

Tutorial: Do. 10:15 - 11:45 in A5,6 - Hilbertraum (C116)

Content: This course will start with basic knowledge of real analysis, includes 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. These are necessary knowledge in modern PDE theories and their applications (for example, in physics, biology and economy).

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