Welcome to this course on Distribution Theory.

I'm taking over this course from Spring 2023, and I'll update the course page and make possibly some changes on the examination format based on discussions with you students - you're welcome to discuss with me your 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 in here  TimeEdit.

We'll meet 3 times; Mondays 13.15-15.00 at MVF21, Tuesdays 10.00-11.45 at MVF26, and Fridays 10:00-11:45 at MVH11.

The meetings will consist of lectures and exercise sessions. Mondays and Tuedays

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 Hasse Carlsson's "Lecture notes on Distributions", Swedish version,   and English version, which can be found on Hasse's webpage. Some exercises from Hörmander's book will be recommended and copies will be handed out.

P-I. Lectures

 

Week	Chapters	Contents
v. 3	1-2	General introduction, test functions, def of distributions, support of a distributions.
v. 4	3-5	Operations on distributions, finite parts, fundamental solutions of some PDE's,
v. 5	6-8	Convergence and convolution of distributions. (Brief introduction on theory of abstract locally convex topological vector spaces on Friday).
v. 6	9-10	Fundamental solutions, Fourier transform
v. 7	11-12	Fourier transform and convolutions.
v. 8	12, 13, 16.	Paley-Wiener's theorem, Fourier series. Poisson summation formula. (Some general abstract theory of locally convex complete vector spaces defined by family of semi-norms.)
v. 9	14, 15	Existence of Fundamental Solutions. Fundamental solutions of elliptic differential operators. Weyl Lemma (Chapt. 8, Thm 8.8). Hypoellipticity of P(D) and zeros of P(x) (Hörmander's theorem; an easier version of Theorem 11.1.3 Ia) in [H, LinPDOp, vol.2, Section 11.1] )
v. 10	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 (it is not correct for |B|=2^n without extra conditions on B).

 

 

P-II. Assignments and Recommended Exercises.

There will be 3 sets of Assignments, to be posted Weeks 4, 6, 7.

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. 

H=Hörmander's book; HC=Hasse Carlsson's lecture notes

Week	Exercises
v.3	HC: 1.2, 1.4, 2.2, 2.3;  H: 3.1.7
v.4	HC: 2.7 (D). 3.1, 3.2, 4.2;  H: 3.1.1, 3.1.2 (D),  3.1.14, 3.1.20a, b, c
v.5	HC: 5.5, 5.6 (D); Continuity of the convolution u\ast \phi in u and \phi (a detailed proof) (D);   H: 3.1.20d,e,f, 3.1.25 (D. Error in Answers/Hints [H, p.399]: No factor of 2.), 3.3.9, 3.3.11, 2.5a
v.6	HC: 8.4 (D on convexity, and relevant question for R^n mainly on the properties of subharmonic functions, \Delta u\ge 0, and difference between convex and subharmonic functions), 8.7;  H: 2.6a, 2.16, 4.2.1, 4.2.2, 4.2.3, 4.2.4
v.7	HC: 4.4, 12.5' (for x^{-m}, H(x), Sgn(x) ) (D), 12.6 (D);     H: 7.1.33, 7.1.35;  HC: 10.8 (D), 12.5(D) (Presenting some general abstract facts about locally convex vector spaces solving 10.8 as an easy consequence, and systematic treatment of f.p;  see the Extra Exercises/Text here, Section 2.4, 2,5)   H: 7.1.6, 7.1.9, 7.1.10, 7.1.11, 7.1.18 (D)
v.8	HC: 12.7, 14.1;  16.4 (Application of Fourier series and Poisson summation formula) (D).  H: 7.1.40 (Expanded in the Extra Exercise) (D). 7.2.7, 7.2.5 (expanded in the Extra Exercise), 7.2.8.
v.9	HC: 14.1, 13.1 (and its variations). Extra Exercise 16 (about Hörmander's theorem) (D) (see the Assignment file).  H: 4.4.6 (D, Image versus co-kernel)
v10	H: 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). 

Exams 2023

2023-March

(Solution)

2023-JUNE

(Solution: page 1, page 2, page 3, page 4)

Some earlier exams:

2021-March 

(Solution)

2019-March

(Solution)

 

 

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