27. Internetseminar "Harmonic Analysis Techniques for Elliptic Operators"

The 27th Internet Seminar on Evolution Equations is entitled "Harmonic Analysis Techniques for Elliptic Operators". The study of the Laplacian on R^n through the Fourier transform lies at the center of classical harmonic analysis. It is Plancherel’s theorem that intimately links the space L^2(R^n) with the theory of weak derivatives and a symbolic calculus for the Laplacian. Examples are the Littlewood–Paley (in)equality, the Riesz transform “estimates”, or the fact that the resolvent (λ − ∆)−1 is given by a nice kernel that yields bounds in L^p(R^n) for p different from 2. Over the last decades, the quest to generalize these properties to elliptic operators Lu = −div(A∇u) with bounded measurable coefficients has triggered the development of new techniques that led to a surge of spectacular results in elliptic and parabolic PDE-theory. We will give an introductory course that covers the cornerstones of this “L-adapted Fourier analysis”. The generalization of the Riesz transform estimates is the famous Kato square root problem whose solution will also be presented in the lectures. A variety of very recent results relying on these techniques will be covered in the project phase.

We expect the participants to have a basic knowledge in functional analysis, bounded operators, foundations of Hilbert spaces and some familiarity with the Fourier transform and functions in one complex variable.

The concept of the “Internet Seminar” originates in 1998 when Rainer Nagel (Tübingen) organized the first Internet Seminar. Since then, many schools have been organized in the same spirit and the experience of the previous editions has shown that these schools are much more effective than traditional schools where participants have a much more passive role. The course is organised in three phases.

• In Phase 1 (October-February), a weekly lecture will be freely accessible over the internet via the ISEM website. The aim of the lectures is to present the theoretic background which lies behind current ongoing research.
• In Phase 2 (April - June), the participants will form small international groups to work on diverse projects which supplement the theory of Phase 1 and provide some applications of it.
• Finally, Phase 3 (17 June - 21 June 2024) consists in a final workshop at the CIRM in Luminy (Marseille, France), where the teams will present their projects and additional lectures will be delivered by leading experts.

ISem team

Moritz Egert (Darmstadt), Robert Haller (Darmstadt), Sylvie Monniaux (Marseille), Patrick Tolksdorf (Karlsruhe)

Website

https://www.mathematik.tu-darmstadt.de/analysis/lehre_analysis/isem27/