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Theory of Differentiation for Advanced Students I - NMAA077

Title:	Teorie derivace pro pokročilé I
Guaranteed by:	Department of Mathematical Analysis (32-KMA)
Faculty:	Faculty of Mathematics and Physics
Actual:	from 2022
Semester:	winter
E-Credits:	3
Hours per week, examination:	winter s.:2/0, Ex [HT]
Capacity:	unlimited
Min. number of students:	unlimited
4EU+:	no
Virtual mobility / capacity:	no
State of the course:	cancelled
Language:	Czech
Teaching methods:	full-time
Teaching methods:	full-time

Classification:	Mathematics > Real and Complex Analysis

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Annotation -

Last update: T_KMA (17.05.2004)

Spaces of weakly differentiable functions (mainly Sobolev spaces), topics in differentiation of functions of several variables important for applied analysis.

Literature - Czech

Last update: T_KMA (17.05.2004)

A. Kufner, O. John, S. Fučík: Function Spaces. Academia, Praha 1977.

J. Lukeš, J. Malý: Míra a integrál, skripta Universita Karlova, Praha 1993.

L. C. Evans, R. E. Gariepy: Measure Theory and Fine Properties of Functions. CRC Press 1992.

E.M. Stein: Singular Integrals and Differentiability Properties of Functions, Princeton 1970.

W.P. Ziemer: Weakly Differentiable Functions. Sobolev Spaces and Function of Bounded Variation, Graduate Text in Mathematics 120, Springer-Verlag 1989.

Syllabus -

Last update: T_KMA (17.05.2004)

1. Covering theorems, maximal operators, Riesz potentials, Lebesgue points

2. Lorentz and Orlicz spaces

3. Mollification

4. Sobolev spaces, density of smooth functions, equivalent definitions.

5. Estimates of a function in terms of Riesz potential of gradient, Poincaré inequality

6. Discusion of continuity and differentiability (approximate, classical)

7. Embedding theorems (sharp)

8. Approximation of Luzin type

9. Traces and extension

10. Fine properties, capacity.