Course content

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.

Learning outcome

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.

Learning methods and activities

Lectures, alternatively supervised self-study.

This course is taught every second year, next time Spring 2026, provided there is a sufficient number of students attending. If there are not enough students, the course may be offered as a supervised self-study.

Course materials

Will be announced at the start of the course.