Department Mathematik



Vorlesung: Mathematisches Seminar: Pseudodifferential operators (SoSe 2020)

[Update 13.04.2020]: Due to the present situation, this lecture course (originally announced as a seminar!)
will be online (uploaded videos, lecture notes). It may move to the class room if the situation improves.
It will start 21 April 2020 (as planned).
For updated information, check back here, on LSF, and on uni2work.
A brief video introduction can be found here.

Hence, most of the information below is obsolete.
Short info: The course is 1 lecture (90min) a week, 3 ECTS, the programme is the same as described below.
There will be an oral exam (whenever that becomes possible again).
It is expected that the book used will be available online very soon.
You will need to sign up at uni2work to get access to the course material.

Time and place:
Tuesday 08:30 – 10:00 in B 251.

First time: 08:30, April 21st (2020) (Introduction and motivation).

The theory of pseudodifferential operators arose in the 1960's as a tool in the study of elliptic partial differential equations (the Laplace equation, Poisson equation, Dirichlet and Neumann boundary value problems etc.). Such operators are a generalisation of Partial Differential Operators (PDO's), and they have since then become a strong and useful tool in many other areas of analysis, such as Harmonic Analysis, Spectral Theory, and Index Theory for elliptic operators on manifolds (they are an important ingredient in many proofs of the Atiyah-Singer Index Theorem).

This seminar will give an elementary introduction to the theory of pseudodifferential operators and their properties. It will include an introduction to the Fourier transform, (tempered) distributions, and Sobolev spaces, which are by themselves very useful tools.

Topics to be discussed: Schwartz functions (S) and tempered distributions (S'), The Fourier transform on S and S', Sobolev spaces, Pseudodifferential symbols, Oscillatory integrals, Pseudodifferential operators (ΨDO's), The action of ΨDO's on S, S', and Sobolev spaces, Global regularity of elliptic PDO's (and ΨDO's), Gårding's inequality, Applications.

3rd year Bachelor students and Master students of Mathematics and Physics, TMP-Master.


Analysis I-III. Basic knowledge of Functional Analysis and/or Partial Differential Equations is helpful, but not required.

English. (Die mündliche Prüfung kann auch auf Deutsch gemacht werden).

[R] X. Saint Raymond, Elementary introduction to the theory of pseudodifferential operators, CRC Press, Boca Raton, 1991. (Available in several copies in the library).

Supplementary literature:
H. Abels, Pseudodifferential and Singular Integral Operators, De Gruyter Textbook, 2012.
S. G. Krantz, Partial Differential Equations and Complex Analysis, CRC Press, Boca Raton, 1992.
M. M. Wong, An Introduction to pseudo-differential Operators, 2nd ed., World Scientific, Singapore, 1999.
B. E. Petersen, Introduction to the Fourier transform & pseudo-differential operators, Pitman, Boston, 1983.
L. Hörmander, The analysis of linear partial differential operators III, Pseudo-Differential Operators, corr. reprint, Springer, Berlin, 2007.
M. Shubin, Pseudodifferential operators and spectral theory, 2nd ed., Springer, Berlin, 2001.
A. Grigis and J. Sjöstrand, Microlocal Analysis for Differential Operators, Cambridge University Press, 1994.

A longer list can be found here.

(For more on Distribution Theory, see
[F-J] F. G. Friedlander and M. Joshi, Introduction to the Theory of Distributions (2nd Edition), Cambridge University Press, 1999. (Available in several copies in the library) - Errata 1 Errata 2.)

Office hours:
Thursday 10:15-11:00 (Room B 408) or by appointment via email.

Datum Title             Remarks        
21.04.2020     Introduction and motivation.
28.04.2020 More motivation on ΨDO's.
05.05.2020 Fourier-transf. & distrib. in R^n I. [R] p.5-10.
12.05.2020 Fourier-transf. & distrib. in R^n II. [R] p.10-15.    
19.05.2020 Sobolev spaces. [R] p.17-23.
26.05.2020 Payley-Wiener-Schwartz Theorem. [R] p.15-17(Thm1.13)+ex.1.7,1.8.
02.06.2020 No lecture. Pentacost/PfingstDienstag.
09.06.2020 Def. and approx. of symbols. [R] p.29-32+ex.2.1,2.2,2.8(a)+(b).
16.06.2020 Oscillatory integrals. [R] p.32-37+ex.2.3,2.4.
23.06.2020 Operations on symbols. [R] p.37-41+ex.2.5,2.8(c)+(d).
30.06.2020 Ellipticity. [R] Thm2.10+ex.2.6+2.9.
07.07.2020 ΨDO's: Action on S and S'. [R] p.47-52.
14.07.2020 Action in Sobolev spaces. [R] p.52-56+ex.3.7(b)(2nd half), maybe ex.3.3,3.4,3.5.
21.07.2020 Gårding's Inequality. [R] Thm3.9+ex.3.7(a)+(b)(1st half).


Letzte Änderung: 03 September 2020 (No more updates)

Thomas Østergaard Sørensen