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Advanced topics from the interpolation theory. Recommended for master students of mathematical analysis.
Last update: T_KMA (02.05.2013)
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Oral exam on a-priori known parts of the course. Last update: Pick Luboš, prof. RNDr., CSc., DSc. (22.07.2018)
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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. Last update: Pick Luboš, prof. RNDr., CSc., DSc. (22.07.2018)
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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.
Last update: Pick Luboš, prof. RNDr., CSc., DSc. (22.07.2018)
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Basic knowledge in measure theory. Last update: Pick Luboš, prof. RNDr., CSc., DSc. (22.07.2018)
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