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19219801 Theory of function spaces and applications

Winter Term 2021/2022

lecture and exercise by Dr. Willem van Zuijlen

Time and place

  • Lecture: Tuesdays 14:00-16:00h.
    During the first (online) meeting we will discuss in which form the lecture will take place. The details for the online-meeting are availabel in Whiteboard. If you do not have access (yet), please send me an email.

  • Final Exam:  Date to be announced in due course. To the wish of the students it will be either written or an oral exam. In consultation with the teacher also a small project/take-home exam can be considered.

Prerequisits: Analysis I — III, Linear Algebra I, II, Functional Analysis.

If you are an FU student you only need to register for the course via CM (Campus Management).
If you are not an FU student, you are required to register via MyCampus/Whiteboard.

Exercises: Every week a new part of lecture notes will come available with excersises.

Course Overview/ Content:

In this course we consider function spaces and more general spaces of distributions, which are generalized functions. The main key in the course is the theory of Harmonic Analysis, which is used to define Besov spaces. Moreover, we treat Bony's para- and resonance products, which allow us to multiply distributions in some cases.

  • Testfunctions and distributions;
  • Convolutions and mollifiers;
  • Fundamental solutions of PDEs;
  • Sobolev spaces;
  • Schwarz functions and tempered distributions;
  • Fourier transform and Fourier multipliers;
  • Besov spaces;
  • Bony para- and resonance products.

The theory of distributions is developed in order to be able to find solutions to (stochastic) partial differential equations, for example when classical solutions do not exist. In particular, the theory of Paracontrolled Calculus (and related the theory of Regularity Structures) is built on techniques that are threatened in this course.

References

During the course lecture notes will be provided, they will be an update from the ones used at the SoSe2020 which can be found on the website of the teacher: https://www.wias-berlin.de/people/vanzuijlen/LN_theory_of_function_spaces.pdf