Teachers: Philippe Jaming and Bernhard Haak
The aim of this course is to provide analytical tools for the study of partial differential equations (PDEs). – 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 aim to introduce tools from operator theory that are useful in control 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.
Operator Theory and Semigroups (M. Tucsnak) – Extrapolation and very weak solutions of linear evolution equations. Applications to wave and heat equations with inhomogeneous boundary conditions. – Holomorphic semigroups: equivalent definitions, characterization of generators, resolvent estimates, fractional powers, maximal regularity, perturbations. – Control and observation operators: admissibility in an abstract framework, applications to systems described by linear PDEs. – Some concepts of controllability and observability in infinite dimensions.