Seminar Nichtlineare Analysis


Sobolev spaces of fractional order


Christof Melcher

Katarzyna E. Mazowiecka

Angewandte Analysis, RWTH Aachen


12. Oktober 2021, 16:00 Uhr per zoom
Meeting ID: 954 1830 2888, Passcode: 987593


Basic knowledge about Sobolev spaces and functional analysis (in particular basic knowledge about the Fourier transform).


As discovered by Gagliardo in 1957, fractional Sobolev spaces appear in the study of traces of Sobolev maps. These spaces are also found in harmonic analysis and in singular integral operators (e.g., Riesz transform). In this course we will learn what the fractional Sobolev spaces are and their basic properties (embeddings, interpolation inequalities, equivalent seminar forms). We will also discover their relation to the Fourier transform and introduce the fractional Laplace operator. We will see how the fractional Laplace operator is related to the Riesz potential. We will also learn that fractional Sobolev spaces are a subset of Besov spaces and introduce the Paley-Littlewood decomposition. If time permits, we will also discuss the paradifferential calculus.

Pace and level of the introductory course will be adjusted to the students. Seminar talks will be embedded. The needs of the students will be followed.


• T. Runst, W. Sickel: Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations

• H. Triebel: Theory of function spaces
• J. Bergh, J. Löfström: Interpolation spaces