The study of the Laplacian through the Fourier transform lies at the center of classical harmonic analysis. It is Plancherels theorem that intimately links square-integrable functions with the theory of weak derivatives and a symbolic calculus for the Laplacian. Examples include LittlewoodPaley inequalities, Riesz transform estimates and CalderónZygmund extrapolation. Over the last decades, the quest to generalize these properties to elliptic operators L in divergence form 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.
Assuming only undergraduate knowledge in analysis and some background on Hilbert spaces and the Fourier transform, the authors develop the cornerstones of this L-adapted Fourier analysis over 14 consecutive lectures. As they delve deeper into the topic, readers make first encounters with maximal functions, Carleson measures and a T(b) theorem. The lectures culminate in a self-contained presentation of the solution to the Kato conjecture, a challenging problem that resisted solution for 40 years until it was finally solved in 2001.
This book can serve as a fully developed curriculum for a first graduate course in harmonic analysis and PDEs. Based on the 27th Internet Seminar on Evolution Equations, organized by the authors in the 2023/24 academic year, each lecture is enriched with original exercises, detailed solutions, and video presentations guiding through each theorems proof and offering additional insights.
Chapter 1. Basics on operator theory.
Chapter
2. Sectorial operators
and sesquilinear forms.
Chapter 3. The form method for elliptic operators.-
Chapter 4. Fourier analysis and the Laplacian.
Chapter 5. Functional
calculus for sectorial operators.
Chapter
6. First applications of
functional calculus.
Chapter
7. H -calculus.
Chapter
8. Quadratic
estimates vs. functional calculus.
Chapter
9. The HardyLittlewood maximal
operator.
Chapter
10. Sobolev embeddings.
Chapter
11. Off-diagonal
behavior.
Chapter
12. Square roots of elliptic operators.
Chapter
13. The
solution of the Kato conjecture: Part I.
Chapter
14. The solution of the
Kato conjecture: Part II.
Moritz Egert is a professor of Mathematics at Technical University of Darmstadt (Germany). He received his PhD in 2015 in Darmstadt under the supervision of Robert Haller and was subsequently Maître de conférences in the Laboratoire de Mathématiques dOrsay at Université Paris-Saclay. He is a specialist in harmonic analysis and partial differential equations. In his research, he combines methods from harmonic analysis, operator theory and geometric measure theory to study partial differential equations in non-smooth settings.
Robert Haller is a lecturer of Mathematics at Technical University of Darmstadt (Germany). In 2003, he recieved his PhD in Darmstadt under the supervision of Matthias Hieber and after a stay at Weierstrass Institute for Applied Analysis and Stochastics in Berlin, he came back to Darmstadt, where he attained his current position in 2009. He is strongly encouraged in teaching mathematics at all levels and pioneered the use of video recordings of lectures in his department. His main topic of research is elliptic and parabolic regularity of partial differential equations in non-smooth situations relying on tools from operator theory, functional analysis and harmonic analysis.
Sylvie Monniaux is professor of Mathematics at Université Aix-Marseille (France). She received her PhD in 1995 in Besançon under the supervision of Wolfgang Arendt on the topic of maximal regularity for parabolic problems. After a 3 year post-doc position at Universität Ulm (Germany), she got a permanent position as Maîtresse de conférences in Marseille. Her research moved towards partial differential equations in non-smooth situations, and more particularly Navier-Stokes equations in rough domains. Her work combines techniques of functional analysis and harmonic analysis.
Patrick Tolksdorf is a lecturer of Mathematics at Karlsruhe Institute of Technology (Germany). He received his PhD in 2016 at the Technical University of Darmstadt under the supervision of Robert Haller and was subsequently a junior professor at Johannes Gutenberg University Mainz. He is a specialist in harmonic analysis and partial differential equations. In his research, he combines methods from harmonic analysis and operator theory to study flow problems, such as the Stokes and Navier-Stokes equations, in non-smooth settings.
