Syllabus
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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.
Annotation
Advanced topics from the interpolation theory. Recommended for master students of mathematical analysis.