Teachers: Philippe Jaming and Bernhard Haak
The aim of this course is to provide analytical tools for the study of partial differential equations (PDEs). The course is split in two parts that are run simultaneously.
– In the first part, we introduce the main tools of harmonic analysis to study the class of singular integral operators.
– In the second part, we give and introduction introduce semigroup theory.
Harmonic Analysis (Ph. Jaming) – $L^p$ and weak $L^p$ spaces, interpolation. – Fourier analysis, Sobolev spaces, Paley-Wiener spaces. – Hardy-Littlewood maximal function: covering lemma, boundedness of the maximal function, application to Lebesgue differentiation theorem. Harmonic functions on the half-space, Poisson kernel and boundary behavior. – Hilbert and Riesz transforms. – Singular integrals, Calderon-Zygmund decomposition. – BMO (bounded mean oscillation). – Littlewood-Paley multiplier theorem and Hörmander multiplier theorem.
Introduction to semigroups (B. Haak) – semigroups and their generators for solving the abstract Cauchy problem: uniformly continuous and strong continuous semigroups, generation theorems (Hille-Yosida and Lumer-Philipps), analytic semigroups, regularity of solutions. At the end of the course we discuss maximal regularity as a tool to solve non-linear equations by fixed point methods. This requires tools from harmonic analysis developed in the first part.