SubjectsSubjects(version: 850)
Course, academic year 2019/2020
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Introduction to Interpolation Theory 2 - NMMA534
Title in English: Úvod do teorie interpolací 2
Guaranteed by: Department of Mathematical Analysis (32-KMA)
Faculty: Faculty of Mathematics and Physics
Actual: from 2019 to 2019
Semester: summer
E-Credits: 4
Hours per week, examination: summer s.:2/0 Ex [hours/week]
Capacity: unlimited
Min. number of students: unlimited
State of the course: not taught
Language: Czech, English
Teaching methods: full-time
Class: M Mgr. MA
M Mgr. MA > Povinně volitelné
Classification: Mathematics > Functional Analysis, Real and Complex Analysis
Incompatibility : NRFA076
Interchangeability : NRFA076
Annotation -
Last update: T_KMA (02.05.2013)
Advanced topics from the interpolation theory. Recommended for master students of mathematical analysis.
Course completion requirements -
Last update: prof. RNDr. Luboš Pick, CSc., DSc. (22.07.2018)

Oral exam on a-priori known parts of the course.

Literature -
Last update: prof. RNDr. Luboš Pick, CSc., DSc. (22.07.2018)

R.A. Adams, Sobolev Spaces,Academic Press, New York, 1975.

C. Bennett, R. Sharpley: Interpolation of Operators, Academic Press, Princeton, 1988.

J. Bergh, J. Löfström: Interpolation Spaces, Springer, Berlin, 1976.

L. Pick, A. Kufner, O. John and S. Fučík: Function Spaces I, De Gruyter, Berlin, 2012.

Syllabus -
Last update: prof. RNDr. Luboš Pick, CSc., DSc. (22.07.2018)
1. Introduction to the interpolation principle

Young functions, Orlicz spaces, Lebesgue spaces, embedding theorems, Minkowski inequality, Hölder inequality, interpolation inequalities in Sobolev embeddings

2. Classical interpolation theorems: Riesz-Thorin convexity theorem

Riesz-Thorin convexity theorem, operator pf strong type, Riesz convexity theorem for positive operators, Hadamard three lines theorem, Riesz-Thorin theorem, Hausdorff-Young inequaity, boundedness of convolution operators on Lebesgue spaces, Hardy inequality, interpolation square

3. Classical interpolation theorems: Yano extrapolation theorem

Integral mean operator, Yano theorem

4. Classical interpolation theorems: Marcinkiewicz theorem

Nonincreasing rearrangement, Lorentz spaces, embeddings, Hölder inequality, Hardy-Littlewood--Pólya principle, operator of weak type, Marcinkiewicz theorem, Hardy-Littlewood maximal operator, Riesz potential, Hilbertov transform, singular integral operators

5. Joint-typeoperators

Calderón operator, Herz inequality, O´Neil inequality, Calderón operator, operator of joint weak type, interpolation of such operators, Lorentz-Zygmund spaces

6. Abstract interpolation theory

Categories and functors, compatible couple, sum and intersection, interpolation space, Aronszajn-Gagliardo theorem

7. Real method of interpolation

Peetre K-functional, Gagliardo completion, Holmstedt formulae, reiteration theorem, J-functional, examples of K-functionals for certain pairs of spaces

8. Interpolation of compact operators

Compact operators on Lebesgue spaces, Cwikel theorem

9. Optimal Sobolev embeddings

Rerrangement-invariant space, Pólya-Szegö inequality, Sobolev space, Sobolev embedding, optimal range construction.

Entry requirements -
Last update: prof. RNDr. Luboš Pick, CSc., DSc. (22.07.2018)

Basic knowledge in measure theory.

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