Department Mathematik



Mathematisches Seminar: Pseudodifferential operators (SoSe 2020)


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


First meeting: 08:30, April 21st (2020) (Intro, motivation, topics).

Vorträge können auch auf Deutsch gehalten werden!

If interested, please sign up via email ( ) until April 17th (2020).

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.

Language: The speakers can choose between English and German.

Literature: [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 (not needed!):
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.

Programme (Talks start at 08:30):
Datum Speaker           Title             Remarks        
21.04.2020     Thomas Sørensen Intro, motivation, topics.
28.04.2020 Thomas Sørensen 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 seminar. 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).

Wie halte ich einen Seminarvortrag? von Prof. Dr. Manfred Lehn, Johannes Gutenberg-Universität Mainz.


Letzte Änderung: 11 Februar 2020.

Thomas Østergaard Sørensen