Vorlesung: Pseudodifferential operators (PsiDO) (SoSe 2021)

[UPDATE 25.02.2021:] Due to the present situation, this course will be online (uploaded videos, lecture notes; all details on uni2work).
It will start 13 April 2021 (as planned).
For updated information, check back here, on LSF, and on uni2work (where all material will be uploaded).
A brief video introduction can be found here.
You will need to sign up (starting April 01) at uni2work to get access to the course material.


Lecture (Vorlesung):
Tue 08-10: Online (videos) and via Zoom; see uni2work.   LSF

Exercises (Übungen):
There are NO exercises!

Synopsis (Kurzbeschreibung):
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 course 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.

Audience (Hörerkreis):
Master students of Mathematics (WP 17.2, 18.1, 18.2, 44.3, 45.2, 45.3), TMP-Master.

Credits:
3 ECTS.

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

Language (Sprache):
English. (Die mündliche Prüfung kan auch auf Deutsch gemacht werden).

Exam (Prüfung):
There will be an oral exam (dates to be announced) (Es wird eine mündliche Prüfung geben).
The exam will be online. See separate webpage (uni2work).

Content (Inhalt):

0. Schwartz functions (S) and tempered distributions (S')

1. The Fourier transform on S and S'

2. Sobolev spaces

3. Pseudodifferential symbols

4. Oscillatory integrals

5. Pseudodifferential operators (ΨDO's)

6. The action of ΨDO's on S, S', and Sobolev spaces

7. Global regularity of elliptic PDO's (and ΨDO's)

8. Gårding's inequality

9. Applications

Literature:
In uni2work you will find a copy of the notes from the lecture.
Above you will find a short description of the content of the lecture.
The lecture will mainly follow the book by X. Saint-Raymond mentioned below.

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:
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:
Via Zoom; see uni2work.

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Letzte Änderung: 28 July 2021 (No more updates).

Thomas Østergaard Sørensen