Differentialoperatorer og Funktionsrum (DifFun)

Welcome to the course on Differential operators and Function spaces, Fall 2011, Block 2. The lectures will be given by Gerd Grubb, and the exercise classes conducted by Kim Petersen.

We shall use the book by Gerd Grubb: "Distributions and Operators", Graduate Texts in Mathematics, Springer Verlag 2009. 

In this course we shall read Chapters 1-3, 12, and 4-6 (in that order), with some omissions, and with consultations of the appendices. 


Course description

Weakly scheduled hours:

Lecture 1 Monday from 15:00 to 17:00 in Auditorium 7
Lecture 2 Wednesday from 10:00 to 12:00 in Auditorium 8
Lecture 3 Wednesday from 15:00 to 16:00 in Auditorium 9
Exercise  class: Wednesday from 13:00 to 15:00 in  1-0-30 (DIKU)

It is customary at the university to start 15 minutes into the hour. So, add 15 min.s to each of the starting times above.

The first lecture begins Monday November 21, 2011 at 15.15 in Aud. 7 at the HCØrsted Institute, Universitetsparken 5. 

Teaching period: November 21 2011 – January 27 2012, in the weeks 47-51 of 2011 and 1-4 of 2012.

There is homework every week, and the homework in weeks 50 of 2011 and 2 of 2012 will be obligatory (counting with 20% each in the final result).  The dates for handing in these two obligatory homeworks are so far planned to be: December 14 and January 11, 2012.  (These dates may be modified if some outer circumstances make it necessary.)

In week 4 2012, on the last day of the course January 25, there will be a written exam of 3 hours counting with 60%. There will be given a grade and there will be an external censor.

You are welcome to collaborate with others in solving the homework problems, but you must write your own formulation of the answer (we do not accept copying). Write in Danish or English.

List of comments to the book: corrections . Additional miscellaneous exercises: miscellaneous


PLAN OF THE COURSE. Past lectures will be described in detail, and tentative plans for future lectures will be posted here. The exercises for each week will be posted here, on or before the Wednesday of the preceding week. Written homework is to be handed in at the exercise session on Wednesdays.



Week 47, Nov. 21 and 23: After explaining the need for generalizations of the concept of a derivative, and mentioning weak derivatives (Chapter 1), we considered the relevant function spaces introduced in Chapter 2.1. First of all, we showed the existence of test functions. Next, we recalled the Banach spaces of continuous or k times continuously differentiable functions on closed intervals in one dimension, closed boxes in n dimensions. Using material from Appendix B, leaving out most proofs, we introduced Fréchet spaces and the important construction of such spaces by use of countable families of seminorms. This was applied to function spaces over open sets and with infinitely many derivatives. Finally, the inductive limit topology on the union of an increasing family of Fréchet spaces was introduced, and applied to the test function space. (From Appendix B we shall in the course mainly use Definition B.4, Th. B.5, Remark B.6, Lemma B.7, Remark B.8, Th. B.9, Th. B.15 and Th. B.16. As for Th. B.18 we use the concrete version Th. 2.5.)

Week 48, Nov. 28 and 30:   The rest of Sections 2.1-2.3 was covered and we began Chapter 3, defining distributions, illustrated by the important examples: locally integrable functions, the delta-measure, and their derivatives. Rules of calculus were covered up until and including Lemma 3.6. (Remark 3.3 was only briefly mentioned, Lemma 3.6 was proved for k=1.)

Week 49, Dec. 5 and 7: We continued Chapter 3, with main topics being the various operations on distributions, the special estimates for distributions with compact support, and the approximation theorems. Lemma 3.6 and Theorem 3.16 were superficially explained, Theorem 3.20 was not mentioned. Section 3.5 was left as voluntary reading. Finally, we began Chapter 12 on unbounded operators, up to and including Theorem 12.5.

Week 50, Dec. 12 and 14: In Chapter 12, the study of unbounded operators in Hilbert space was covered essentially until the end of Sections 12.4. We skipped Theorems 12.11 and 12.12, and the considerations on uniqueness in Corollaries 12.22 and 12.25 were only lightly touched. Corollary 12.21 was postponed, Remark 12.23 skipped. This was a week with obligatory exercises (handed in on December 14).

Week 51, Dec. 19 and 21: Back to distribution theory, Chapter 4. On December 19, the topic was realizations of differential operators and Sobolev spaces, Sections 4.1-4.2 (some rules were given only in survey form). Because of travels, the lectures 10-12 on December 21 were given by Heiko Gimperlein, on Section 4.3. Lectures 15-16 were postponed into the new year, as an extra hour 16-17 on January 4.

Week 1 (2012), Jan. 2 and 4: The lectures on January 2 were given by Kim Petersen, on Section 4.4 until Theorem 4.29. On January 4 (lectures 10-12 and 15-17), we ended Section 4.4 and did Ch. 5, Sections 5.1-5.3, on basic properties of the Fourier transformation for temperate distributions.

Week 2, Jan. 9 and 11: Chapter 6, Section 6.1-6.3, on x-independent pseudodifferential operators, Sobolev spaces over R^n of all orders, Sobolev’s Theorem and the Structure Theorem. This was a week with obligatory exercises (handed in on January 11).

Week 3, Jan. 16 and 18: Sections 6.4 and 5.6. Lectures planned for Monday 15-17 and Wednesday 10-12. Exercises on Wednesday used all three hours 13-16. The lectures Wednesday were cancelled because of illness.

Week 4, Jan. 23 and 25:  Question session (also with exercises) on January 23, written exam January 25.



Week 47:  We use this first session to go through Appendix A, refreshing some known background facts and getting acquainted with others. Do the exercises A.2 (a) (and (b) if there is time) and A.3 in class. 

Week 48:  2.2, 2.3, 2.4, 2.5, B.3, B.10, B.11, B.12. The underlined exercises are to be answered in written form.

Week 49:  2.6, 2.7, 3.1, 3.2(a), 3.3, 3.4, 3.5, 3.8, 3.9.

Week 50:  B.13, 3.7, 3.10, 3.11, 3.12, 3.13, 3.17, 6.9, 6.31. The underlined exercises are obligatory, to be handed in at the exercise class on December 14, 2011.

Week 51:  12.1, 12.3, 12.4, 12.8, 12.10, 12.12, 12.23, 12.35, 6.39 (this is posted in miscellaneous ).

Week 1 (2012):  4.2, 4.3, 4.4, 4.7, 4.10, 4.11, 4.14, 4.23, 4.24. In Exercise 4.14, you can use the outcome of Remark 4.21 without further explanation.

Week 2: 4.9, 4.25, 5.1, 5.3, 5.4(a), 5.6, 5.7, 5.10, 6.36, 6.41. The underlined exercises are obligatory, to be handed in at the exercise class on January 11, 2012. In Exercise 6.36, you can use the results of Exercise 3.9.

Week 3: 4.13, 5.8, 6.1, 6.4, 6.5, 6.6(a,c), 6.22, 6.26, 6.28, 6.38. Exercise class from 13h to 16h.

Week 4: Written exam.


Old examination sets: The exercises 6.33-35 were a 2-hour test in April 2006, 6.36-38 were a 3-hour test in April 2007, 6.41-43 were a 2-hour test in April 2009, and 6.44-46 were a 3-hour test in January 2011. Most of the exercises in Chapter 6 (from 6.7 on) are old exam problems, but the course was larger in earlier versions, and exams were often take-home tests.


Eksamen finder sted på CSS (Øster Farimagsgade 5) i lokale 7-01-30 i bygning 7 kl. 9-12. Se evt. oversigtskort på


Medbring og aflever de obligatoriske sæt, så de kan forelægges for censor ved den endelige bedømmelse.

Kære alle, her er omsider resultatet af eksamen (med afleveringssættene vægtet 20% hver og slutprøven vægtet 60%)

Nr. 1: 10.  Nr. 2: 10. Nr. 3: 02. Nr. 4: 7. Nr. 5: 02. Nr. 9: 4.

Nr. 11: 7. Nr. 12: 7. Nr. 14: 12. Nr. 15: 10. Nr. 16: 7. Nr. 17: 4.

Det var godt at alle bestod. Jeg takker for jeres indsats med kurset, og ønsker jer held og lykke videre i studiet.

Venlig hilsen, Gerd.




Pensum er baseret på Chapters 1-6, Chapter 12 og Appendices i bogen ”Distributions and Operators”.


Betegnelsen "kursorisk" drejer sig om emner, der har været gennemgået overfladisk eller nævnt uden bevis (som man altså ikke skal stå til regnskab for), men hvor resultatet kan bruges i senere tekst og i øvelser.


Alle tekststykker betegnet ved Remark (indenfor nedennævnte afsnit) er kursoriske.


Ch. 1: Kursorisk.


Ch. 2: Sections 2.1-2.3 indgår med fuld værdi, Section 2.4 er kursorisk.


Ch. 3: Sections 3.1-3.4 indtil Th. 3.20 (og exklusive denne sætning). Her følgende læst kursorisk: Lemma 3.6, Th. 3.14, Th. 3.16.


Ch. 4: Sections 4.1-4.4, med kursoriske dele: Th. 4.9, Th. 4.10 vedrørende R^n_+ og Omega, Cor. 4.11 vedrørende Omega, Th. 4.12 og Th. 4.25.


Ch. 5: Sections 5.1-5.3, med kursoriske dele: Th. 5.4, Th. 5.5 og Th. 5.6 (da de er forudsat bekendt).


Ch. 6: Sections 6.1-6.4, med overspringelse af Lemma 6.7 og beviset for ikke-hele s i Lemma 6.17 2^o.


Ch. 12: Sections 12.1-12.5, med overspringelse af Th. 12.11 og Th. 12.12; her er Cor. 12.22 og 12.25 kursoriske.


App. A er baggrundsviden. App. B er læst kursorisk, og vi bruger heraf især Definition B.4, Th. B.5, Remark B.6, Lemma B.7, Remark B.8, Th. B.9, Th. B.15 og Th. B.16. App. C medtages kursorisk.