Department Mathematik



Website for the lecture

Pseudodifferential operators

SoSe 2016

Pseudodifferentialoperatoren mit Übungen

The next (and last) lecture will take place on wednesday, 06.07.2016.


The exam will take place on July 18. Please sign up by email (until monday, 11.07.2016), including the following information: Name, Matrikelnummer, Studiengang, HF/NF, Abschluss, Buchungsstelle and whether you wish to be examined in German or in English. You may also indicate a preferred time slot.


Pseudodifferential operators are generalizations of partial differential operators and were introduced into analysis in the 1960s as a tool in the study of elliptic partial differential equations. In quantum mechanics, they were used even earlier, in connection with the quantization procedure. This course is an introduction to the global theory of pseudodifferential operators on Euclidean space. Topics to be covered include: Fourier transform and tempered distributions, pseudodifferential symbols and asymptotic expansion, adjoints and products of pseudodifferential operators, parametrix construction for elliptic pseudodifferential operators, elliptic regularity estimates and more advanced topics.


Master students mathematics, TMP master.


Analysis I-III. A nodding acquaintance with functional analysis and some familiarity with the Fourier transform are helpful.

Time and Place:

Mo 16-18, B039 (lecture), We 12-14, B039 (exercises).

Primary literature:

H. Abels, Pseudodifferential operators and singuar integral operators: an introduction with applications, De Gruyter textbook, 2012. (see also this link for an online version of lecture notes by the same author.)
X. Saint Raymond, Elementary introduction to the theory of pseudodifferential operators, CRC Press, Boca Raton, 1991.

Secondary literature:

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, 1994.
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.
For a longer list of references, see also this link.

Exercise sheets:

I will update the exercise sheets if there were typos! Copies of my handwritten solutions are available by request (E-Mail).

Exercise sheet 1

Exercise sheet 2

Exercise sheet 3

Exercise sheet 4

Exercise sheet 5

Exercise sheet 6

Exercise sheet 7

Solution sheet 7, Exercises 3 and 5

Exercise sheet 8

Exercise sheet 9

Exercise sheet 10