Harmonic Analysis

The course will treat central concepts and results in modern harmonic analysis, which are developments from classical Fourier analysis. One possible theme may be harmonic analysis related to the study of singular integrals and complex and real methods. Some key concepts are: maximal functions, Calderon-Zygmund decompositions, the Hilbert transform, Littlewood-Paley theory, Hardy spaces, Carleson measures, Cauchy integrals, singular integral operators. The course may also cover a more abstract direction dealing with generalizations of classical Fourier analysis from the unit circle to locally compact Abelien groups. Key concepts are then Haar measure, convolution, the dual group and Fourier transformation, positive definite functions, the inversion theorem, Plancherel's theorem, Pontryagin's duality theorem, the Bohr compactification.

1. Knowledge. The course will treat central concepts and results in modern harmonic analysis, which are developments from classical Fourier analysis. One possible theme may be harmonic analysis related to the study of singular integrals and complex and real methods. Some key concepts are: maximal functions, Calderon-Zygmund decompositions, the Hilbert transform, Littlewood-Paley theory, Hardy spaces, Carleson measures, Cauchy integrals, singular integral operators. The course may also cover a more abstract direction dealing with generalizations of classical Fourier analysis from the unit circle to locally compact Abelien groups. Key concepts are then Haar measure, convolution, the dual group and Fourier transformation, positive definite functions, the inversion theorem, Plancherel's theorem, Pontryagin's duality theorem, the Bohr compactification. 2. Skills. The students should learn the basics of the contemporary Harmonic Analysis and be able to apply its methods in related subjects of Mathematics. 3. Competence. The students should be able to participate in scientific discussions and conduct research at a high international level in contemporary and classical Harmonic analysis as well as its applications to various areas of Mathematics.

Lectures, alternatively supervised self-study.

The course will be taught as needed. If there are few PhD students, the course is only given as a guided self-study.