Course syllabus
Welcome to this course on Distribution Theory, VT2025.
I'm taking over this course since Spring 2023. I'll try to make some changes on (a) the contents of the course trying to bring some recent development related to distribution theory, (b) bring more exercises apart from those in Hörmander's book (see below). You students are welcome to make suggestions about the course, lectures, lecture notes, exercises and to share with us your study techniques and experience of learning advanced courses.
This page contains the program of the course: lectures, exercise sessions. Other information, such as learning outcomes, teachers, literature and examination, are in a separate course PM.
Program
The schedule of the course is
Week 4-11; Mon., 13:15-15, MVF21,
Tues., 10-11:45, MVF26,
Fri.. 10-11:45, MVH11.
Mondays and Tuesdays are lectures. Fridays will be exercises and lectures combined; we try to solve those exercises marked with (D) in the list of recommended exercises below; we start with group discussions and eventually summarize some solutions.
As literature we will use
[HC] Hasse Carlsson, Lecture Notes on Distributions,the updated English 2025Jan-version).
The lecture notes were based on an earlier lecture note of and the volume 1 of the four-volume book series of
[LH] Lars Hörmander, The Analysis of Linear Partial Differential Operators, Springer-Verlag; Vol. 1, Ch. I-IV, Ch. VII, 7.1-7.2. (Ch. 9, 14, 15 in [HC] about fundamental solutions and hypoelliptic operators are not in [LH], and there is a more theory of hypoelliptic operators in Hörmander's book, Vol. 2, Chapter XI.)
Chalmers library has a digital online version of the Hörmander's book.
(Go to the library and seek the book with the full title).
The following book contains more detailed proofs.
W. Donoghue, Distributions and Fourier Transform. Academic Press.
P-I. Lectures
| Week | Chapters | Contents |
| W. 4 | 1-2 | General introduction, test functions, various spaces of function spaces, def of distributions, support of a distributions |
| W. 5 | 3-4 | Operations on distributions, finite parts. fundamental solutions of some PDE's |
| W. 6 | 5-8 | fundamental solutions of some PDE's. Convergence and convolution of distributions. (This is called weak*-convergence in Functional Analysis.). |
| W. 7 | 9-10 | Fundamental solutions, Fourier transform and its application in solving PDEs. |
| W. 8 | 11-12 | Fourier transform and convolution of test functions and (tempered) distributions |
| W. 9 | 13, 16. |
Paley-Wiener's theorem, Fourier series. Poisson summation formula. |
| W. 10 | 14, 15 |
Existence of Fundamental Solutions. Fundamental solutions of elliptic differential operators. |
| W. 11 | 15, 17 | Application: (Brief account on) Mean-value properties of harmonic functions, Heisenberg uncertainty principle, Sobolev inequalities; (Detailed proof of) Minkowski's theorem for |B|>2^n. |
P-II. Assignments and Recommended Exercises.
There will be 3 sets of Assignments, to be posted Weeks 5, 7, 9 (study week 2, 4, 6) and to be handed in within two weeks. They are found under the Assignment Section (on the left column of this homepage.)
Each submission will be graded with U/G,/VG. Submissions will grade U have to be re-submitted to get G.
There will be a written exam. To get G, both Assignments and Exam have to be G or above VG. To get VG both Assignments and Exam have to be VG.
LH=Hörmander's book; HC=Hasse Carlsson's lecture notes.
We shall handout some exercises from LH.
(D) below are exerecise for Discussions/Demonstrations on Fridays.
| Week | Exercises | Some Solutions (posted after our discussions Friday Morning) |
|---|---|---|
| W.4 |
HC: 1.2, 1.4, 2.2, 2.3; LH: 3.1.7. |
|
| W.5 |
HC: 2.7 (D). 3.1, 3.2, 4.2; LH: 3.1.1, 3.1.2 (D), 3.1.14, 3.1.20a, b, c. Extra Ex. 1(d) (D.) (Hints on) LH: 2.2 (D), 3.1.5, 3.1.6 (these three are related) Exercise about finite part of functions x^\alpha. Solution of Exercise 2.7. (Hint: Use the idea of the proof of Thm 2.13 and the result of Thm 2.14) |
|
| W.6 |
HC: 5.5, 5.6 (D); Extra Ex on continuity of u\ast \phi; and Ex. on elementary proof on fundamental solution (D). LH: 2.5a, 2.6 (b) (D), 3.1.20 d, e, f, 3.1.25 (D. Error in Answers/Hints [LH, p.399]: No factor of 2.), 3.3.9, 3.3.11, |
|
| W.7 |
HC: 8.4 (D), 8.7; LH: 2.16, 4.1.1, 4.2.1, 4.2.2, 4.2.3, 4.2.4, 4.2.8 (D). |
|
| W.8 |
HC: 4.4, 12.5' (for x^{-m}, H(x), Sgn(x) , 12.6 (D); LH: 7.1.33, 7.1.35; HC: 10.8, 12.5(D) (systematic treatment of f.p as distribution is treated during the lecture in Week 5.) LH: 7.1.6, 7.1.9, 7.1.10, 7.1.11, 7.1.18. |
|
| W.9 |
HC: 12.7, 14.1; 16.4 (Application of Fourier series and Poisson summation formula) (D). LH: 7.1.36 (for one dimension and for general \alpha)(D). 7.1.40. 7.2.7, 7.2.5, 7.2.8, 7.2.9, 7.2.10(D-brief discussions). |
|
| W.10 |
HC: 14.1, 13.1 (and its variations). Extra Exercise 1(d)(f) (about hypoellipticity, ellipticity, strong ellipticity) (D). LH: 4.4.6 (D, Image versus co-kernel), 7.1.3 (D, this is a discrete version of 7.1.5 and our homework) |
|
| W.11 | LH: 7.1.26, 7.6.4 (D) |
P-III. List of Definitions and Theorems.
Here is a list of the main Def's/Thm's, one of the examination problem will be about to formulate them and related theoretical questions.
P-IV. Examination.
Written examination in the middle of March with 4 problems and one on statements/definitions and related theoretical question (sufficiency and necessity, examples and counter examples, differences between concepts).
Exam 2025: Questions and Solutions.
Exam 2025 June: Questions and Solutions.
Exams 2023
(Solution: page 1, page 2, page 3, page 4)
Some earlier exams:
Course summary:
| Date | Details | Due |
|---|---|---|